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{{More footnotes|date=April 2010}}
In [[universal algebra]] and in [[model theory]], a '''structure''' consists of a [[Set (mathematics)|set]] along with a collection of [[finitary|finitary operations]], and [[finitary relation|relation]]s that are defined on it.
 
Universal algebra studies structures that generalize the [[algebraic structure]]s such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]], [[field (mathematics)|fields]] and [[vector space]]s. The term '''universal algebra''' is used for structures with no [[relation symbol]]s.<ref>Some authors refer to structures as "algebras" when generalizing universal algebra to allow [[relation (mathematics)|relations]] as well as functions.</ref>
 
Model theory has a different scope that encompasses more arbitrary theories, including [[foundations of mathematics|foundational]] structures such as models of [[set theory]]. From the model-theoretic point of view, structures are the objects used to define the semantics of [[first-order logic]]. For a given theory in model theory, a structure is called a '''model''', if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a ''semantic model'' when one discusses the notion in the more general setting of [[mathematical model]]s. Logicians sometimes refer to structures as [[interpretation (logic)|interpretation]]s.<ref>{{cite book|editor=Anthonie Meijers|title=Philosophy of technology and engineering sciences|year=2009|publisher=Elsevier|isbn=978-0-444-51667-1|series=Handbook of the Philosophy of Science|volume=9|author=Wilfrid Hodges|chapter=Functional Modelling and Mathematical Models}}</ref>
 
In [[database theory]], structures with no functions are studied as models for relational [[databases]], in the form of [[relational model]]s.
 
== Definition ==
{{see also|Model theory#Universal algebra|Universal algebra#Basic idea}}
Formally, a '''structure''' can be defined as a triple <math>\mathcal A=(A, \sigma, I)</math> consisting of a '''domain''' ''A'', a [[signature (logic)|signature]] σ, and an '''interpretation function''' ''I'' that indicates how the signature is to be interpreted on the domain. To indicate that a structure has a particular signature σ one can refer to it as a σ-structure.
 
=== Domain ===
The [[domain of discourse|domain]] of a structure is an arbitrary set; it is also called the '''underlying set''' of the structure, its '''carrier''' (especially in universal algebra), or its '''universe''' (especially in model theory). In classical first-order logic, the definition of a structure prohibits the [[empty domain]].<ref>This is similar to the definition of a [[prime number]] in elementary [[number theory]], which has been carefully chosen so that the [[irreducible element|irreducible]] number 1 is not considered prime. The convention that the domain of a structure may not be empty is particularly important in logic, because several common inference rules, notably, [[universal instantiation]], are not sound when empty structures are permitted. A logical system that allows the empty domain is known as an [[free logic|inclusive logic]].</ref>
 
Sometimes the notation <math>\operatorname{dom}(\mathcal A)</math> or <math>|\mathcal A|</math> is used for the domain of <math>\mathcal A</math>, but often no notational distinction is made between a structure and its domain. (I.e. the same symbol <math>\mathcal A</math> refers both to the structure and its domain.)<ref>As a consequence of these conventions, the notation <math>|\mathcal A|</math> may also be used to refer to the [[cardinality]] of the domain of <math>\mathcal A</math>. In practice this never leads to confusion.</ref>
 
=== Signature ===
{{main|Signature (logic)}}
The [[signature (logic)|signature]] of a structure consists of a set of '''function symbols''' and '''relation symbols''' along with a function that ascribes to each symbol ''s'' a [[natural number]] <math>n=\operatorname{ar}(s)</math> which is called the '''arity''' of ''s'' because it is the [[arity]] of the interpretation of ''s''.
 
Since the signatures that arise in [[algebra]] often contain only function symbols, a signature with no relation symbols is called an '''algebraic signature'''. A structure with such a signature is also called an '''algebra'''; this should not be confused with the notion of an [[algebra over a field]].
 
=== Interpretation function ===
{{main|Interpretation (model theory)}}
The '''interpretation function''' ''I'' of <math>\mathcal A</math> assigns functions and relations to the symbols of the signature. Each function symbol ''f'' of arity ''n'' is assigned an [[arity|''n''-ary]] function <math>f^{\mathcal A}=I(f)</math> on the domain. Each relation symbol ''R'' of arity ''n'' is assigned an ''n''-ary relation <math>R^{\mathcal A}=I(R)\subseteq A^{\operatorname{ar(R)}}</math> on the domain. A nullary function symbol ''c'' is called a '''constant symbol''', because its interpretation ''I(c)'' can be identified with a constant element of the domain.
 
When a structure (and hence an interpretation function) is given by context, no notational distinction is made between a symbol ''s'' and its interpretation ''I(s)''. For example if ''f'' is a binary function symbol of <math>\mathcal A</math>, one simply writes <math>f:\mathcal A^2\rightarrow\mathcal A</math> rather than <math>f^{\mathcal A}:|\mathcal A|^2\rightarrow|\mathcal A|</math>.
 
=== Examples ===
The standard signature σ<sub>''f''</sub> for [[field (mathematics)|fields]] consists of two binary function symbols + and ×, a unary function symbol '''−''', and the two constant symbols 0 and 1.
Thus a structure (algebra) for this signature consists of a set of elements ''A'' together with two binary functions, a unary function, and two distinguished elements; but there is no requirement that it satisfy any of the field axioms. The [[rational number]]s ''Q'', the [[real number]]s ''R'' and the [[complex number]]s ''C'', like any other field, can be regarded as σ-structures in an obvious way:
 
::<math>\mathcal Q = (Q, \sigma_f, I_{\mathcal Q})</math>
::<math>\mathcal R = (R, \sigma_f, I_{\mathcal R})</math>
::<math>\mathcal C = (C, \sigma_f, I_{\mathcal C})</math>
 
where
 
::<math>I_{\mathcal Q}(+)\colon Q\times Q\to Q</math> is addition of rational numbers,
::<math>I_{\mathcal Q}(\times)\colon Q\times Q\to Q</math> is multiplication of rational numbers,
::<math>I_{\mathcal Q}(-)\colon Q\to Q</math> is the function that takes each rational number ''x'' to -''x'', and
::<math>I_{\mathcal Q}(0)\in Q</math> is the number 0 and
::<math>I_{\mathcal Q}(1)\in Q</math> is the number 1;
and <math>I_{\mathcal R}</math> and <math>I_{\mathcal C}</math> are similarly defined.
 
But the ring ''Z'' of [[integer]]s, which is not a field, is also a σ<sub>''f''</sub>-structure in the same way. In fact, there is no requirement that ''any'' of the field axioms hold in a σ<sub>''f''</sub>-structure.
 
A signature for [[ordered field]]s needs an additional binary relation such as &lt; or ≤, and therefore structures for such a signature are not algebras, even though they are of course [[algebraic structure]]s in the usual, loose sense of the word.
 
The ordinary signature for set theory includes a single binary relation ∈.  A structure for this signature consists of a set of elements and an interpretation of the ∈ relation as a binary relation on these elements.
 
== Induced substructures and closed subsets ==
<math>\mathcal A</math> is called an [[substructure|(induced) substructure]] of <math>\mathcal B</math> if
* <math>\mathcal A</math> and <math>\mathcal B</math> have the same signature <math>\sigma(\mathcal A)=\sigma(\mathcal B)</math>;
* the domain of <math>\mathcal A</math> is contained in the domain of <math>\mathcal B</math>: <math>|\mathcal A|\subseteq |\mathcal B|</math>; and
* the interpretations of all function and relation symbols agree on <math>|\mathcal B|</math>.
 
The usual notation for this relation is <math>\mathcal A\subseteq\mathcal B</math>.
 
A subset <math>B\subseteq|\mathcal A|</math> of the domain of a structure <math>\mathcal A</math> is called '''closed''' if it is closed under the functions of <math>\mathcal A</math>, i.e. if the following condition is satisfied:  for every natural number ''n'', every ''n''-ary function symbol ''f'' (in the signature of <math>\mathcal A</math>) and all elements <math>b_1,b_2,\dots,b_n\in B</math>, the result of applying ''f'' to the ''n''-tuple <math>b_1b_2\dots b_n</math> is again an element of ''B'': <math>f(b_1,b_2,\dots,b_n)\in B</math>.
 
For every subset <math>B\subseteq|\mathcal A|</math> there is a smallest closed subset of <math>|\mathcal A|</math> that contains ''B''. It is called the closed subset '''generated''' by ''B'', or the '''hull''' of ''B'', and denoted by <math>\langle B\rangle</math> or <math>\langle B\rangle_{\mathcal A}</math>. The operator <math>\langle\rangle</math> is a [[finitary closure operator]] on the [[power set|set of subsets]] of <math>|\mathcal A|</math>.
 
If <math>\mathcal A=(A,\sigma,I)</math> and <math>B\subseteq A</math> is a closed subset, then <math>(B,\sigma,I')</math> is an induced substructure of <math>\mathcal A</math>, where <math>I'</math> assigns to every symbol of σ the restriction to ''B'' of its interpretation in <math>\mathcal A</math>. Conversely, the domain of an induced substructure is a closed subset.
 
The closed subsets (or induced substructures) of a structure form a [[lattice (order)|lattice]]. The [[meet (mathematics)|meet]] of two subsets is their intersection. The [[join (mathematics)|join]] of two subsets is the closed subset generated by their union. Universal algebra studies the lattice of substructures of a structure in detail.
 
=== Examples ===
Let σ&nbsp;=&nbsp;{+,&nbsp;×,&nbsp;−,&nbsp;0,&nbsp;1} be again the standard signature for fields. When regarded as σ-structures in the natural way, the [[rational number]]s form a substructure of the [[real number]]s, and the real numbers form a substructure of the [[complex number]]s. The rational numbers are the smallest substructure of the real (or complex) numbers that also satisfies the field axioms.
 
The set of integers gives an even smaller substructure of the real numbers which is not a field. Indeed, the integers are the substructure of the real numbers generated by the empty set, using this signature.  The notion in abstract algebra that corresponds to a substructure of a field, in this signature, is that of a [[subring]], rather than that of a [[subfield]].
 
The most obvious way to define a [[graph (mathematics)|graph]] is a structure with a signature σ consisting of a single binary relation symbol ''E''. The vertices of the graph form the domain of the structure, and for two vertices ''a'' and ''b'', <math>(a,b)\!\in \text{E}</math>&nbsp; means that ''a'' and ''b'' are connected by an edge. In this encoding, the notion of induced substructure is more restrictive than the notion of [[Glossary of graph theory#Subgraphs|subgraph]]. For example, let ''G'' be a graph consisting of two vertices connected by an edge, and let ''H'' be the graph consisting of the same vertices but no edges. ''H'' is a subgraph of ''G'', but not an induced substructure. The notion in [[graph theory]] that corresponds to induced substructures is that of induced subgraphs.
 
== Homomorphisms and embeddings ==
{{see also|Universal algebra#Basic constructions}}
 
=== Homomorphisms ===
Given two structures <math>\mathcal A</math> and <math>\mathcal B</math> of the same signature σ, a '''(σ-)homomorphism''' from <math>\mathcal A</math> to <math>\mathcal B</math> is a [[map (mathematics)|map]] <math>h:|\mathcal A|\rightarrow|\mathcal B|</math> that preserves the functions and relations. More precisely:
 
* For every ''n''-ary function symbol ''f'' of σ and any elements <math>a_1,a_2,\dots,a_n\in|\mathcal A|</math>, the following equation holds:
::<math>h(f(a_1,a_2,\dots,a_n))=f(h(a_1),h(a_2),\dots,h(a_n))</math>.
* For every ''n''-ary relation symbol ''R'' of σ and any elements <math>a_1,a_2,\dots,a_n\in|\mathcal A|</math>, the following implication holds:
::<math>(a_1,a_2,\dots,a_n)\in R \implies (h(a_1),h(a_2),\dots,h(a_n))\in R</math>.
 
The notation for a homomorphism ''h'' from <math>\mathcal A</math> to <math>\mathcal B</math> is <math>h: \mathcal A\rightarrow\mathcal B</math>.
 
For every signature σ there is a [[concrete category|concrete]] [[category (mathematics)|category]] σ-'''Hom''' which has σ-structures as objects and σ-homomorphisms as [[morphism (category theory)|morphisms]].
 
A homomorphism <math>h: \mathcal A\rightarrow\mathcal B</math> is sometimes called '''strong''' if for every ''n''-ary relation symbol ''R'' and any elements <math>b_1,b_2,\dots,b_n\in|\mathcal B|</math> such that <math>(b_1,b_2,\dots,b_n)\in R</math>, there are <math>a_1,a_2,\dots,a_n\in|\mathcal A|</math> such that <math>(a_1,a_2,\dots,a_n)\in R</math> and <math>b_1=h(a_1),\,b_2=h(a_2),\,\dots,\,b_n=h(a_n).</math>
The strong homomorphisms give rise to a subcategory of σ-'''Hom'''.
 
=== Embeddings ===
A (σ-)homomorphism <math>h:\mathcal A\rightarrow\mathcal B</math> is called a (σ-)'''embedding''' if it is [[injective function|one-to-one]] and
 
* for every ''n''-ary relation symbol ''R'' of σ and any elements <math>a_1,a_2,\dots,a_n</math>, the following equivalence holds:
::<math>(a_1,a_2,\dots,a_n)\in R \iff(h(a_1),h(a_2),\dots,h(a_n))\in R</math>.
 
Thus an embedding is the same thing as a strong homomorphism which is one-to-one.
The category σ-'''Emb''' of σ-structures and σ-embeddings is a concrete [[subcategory]] of σ-'''Hom'''.
 
Induced substructures correspond to [[subobject]]s in σ-'''Emb'''. If σ has only function symbols, σ-'''Emb''' is the subcategory of [[monomorphism]]s of σ-'''Hom'''. In this case induced substructures also correspond to subobjects in σ-'''Hom'''.
 
=== Example ===
As seen above, in the standard encoding of graphs as structures the induced substructures are precisely the induced subgraphs. However, a [[graph homomorphism|homomorphism between graphs]] is the same thing as a homomorphism between the two structures coding the graph. In the example of the previous section, even though the subgraph ''H'' of ''G'' is not induced, the identity map id:&nbsp;''H''&nbsp;→&nbsp;''G'' is a homomorphism. This map is in fact a [[monomorphism]] in the category σ-'''Hom''', and therefore ''H'' is a [[subobject]] of ''G'' which is not an induced substructure.
 
=== Homomorphism problem ===
The following problem is known as the ''homomorphism problem'':
 
:Given two finite structures <math>\mathcal A</math> and <math>\mathcal B</math> of a finite relational signature, find a homomorphism <math>h:\mathcal A\rightarrow\mathcal B</math> or show that no such homomorphism exists.
 
Every [[constraint satisfaction problem]] (CSP) has a translation into the homomorphism problem.<ref>{{Citation | last=Jeavons | first=Peter | coauthors=David Cohen; Justin Pearson | year=1998 | title=Constraints and universal algebra | journal=Annals of Mathematics and Artificial Intelligence | volume=24 | pages=51–67 | doi=10.1023/A:1018941030227 | postscript=. }}</ref>
Therefore the [[Complexity of constraint satisfaction#Constraint satisfaction and the homomorphism problem|complexity of CSP]] can be studied using the methods of [[finite model theory]].
 
Another application is in [[database theory]], where a [[relational model]] of a [[database]] is essentially the same thing as a relational structure. It turns out that a [[conjunctive query]] on a database can be described by another structure in the same signature as the database model. A homomorphism from the relational model to the structure representing the query is the same thing as a solution to the query. This shows that the conjunctive query problem is also equivalent to the homomorphism problem.
 
== Structures and first-order logic ==
{{see also|Model theory#First-order logic|Model theory#Axiomatizability, elimination of quantifiers, and model-completeness}}
Structures are sometimes referred to as "first-order structures". This is misleading, as nothing in their definition ties them to any specific logic, and in fact they are suitable as semantic objects both for very restricted fragments of first-order logic such as that used in universal algebra, and for [[second-order logic]]. In connection with first-order logic and model theory, structures are often called '''models''', even when the question "models of what?" has no obvious answer.
 
=== Satisfaction relation ===
Each first-order structure <math>\mathcal{M}</math> has a '''satisfaction relation''' <math>\mathcal{M} \vDash \phi</math> defined for all formulas <math>\, \phi</math>  in the language consisting of the language of <math>\mathcal{M}</math> together with a constant symbol for each element of ''M'', which is interpreted as that element.
This relation is defined inductively using Tarski's [[T-schema]].
 
A structure <math>\mathcal{M}</math> is said to be a '''model''' of a [[Theory (mathematical logic)|theory]] ''T'' if the language of <math>\mathcal{M}</math> is the same as the language of ''T'' and every sentence in ''T'' is satisfied by <math>\mathcal{M}</math>.    Thus, for example, a "ring" is a structure for the language of rings that satisfies each of the ring axioms, and a model of [[Zermelo–Fraenkel axioms|ZFC set theory]] is a structure in the language of set theory that satisfies each of the ZFC axioms.
 
=== Definable relations ===
An ''n''-ary relation ''R'' on the universe ''M'' of a structure <math>\mathcal{M}</math> is said to be '''definable''' (or '''explicitly definable''', or <math>\emptyset</math>-'''definable''') if there is a formula φ(''x''<sub>1</sub>,...,''x''<sub>''n''</sub>) such that
 
:<math>R = \{ (a_1,\ldots,a_n ) \in M^n : \mathcal{M} \vDash \phi(a_1,\ldots,a_n)\}.</math>
In other words, ''R'' is definable if and only if there is a formula φ such that
 
:<math>(a_1,\ldots,a_n ) \in R \Leftrightarrow  \mathcal{M} \vDash \phi(a_1,\ldots,a_n)</math>
is correct.
 
An important special case is the definability of specific elements. An element ''m'' of ''M'' is definable in <math>\mathcal{M}</math> if and only if there is a formula φ(''x'') such that
 
:<math>\mathcal{M}\vDash \forall x ( x = m \leftrightarrow \phi(x)).</math>
 
====Definability with parameters====
A relation ''R'' is said to be '''definable with parameters''' (or <math>|\mathcal M|</math>-'''definable''') if there is a formula φ with parameters from <math>\mathcal{M}</math> such that ''R'' is definable using φ.    Every element of a structure is definable using the element itself as a parameter.
 
It should be noted that some authors use ''definable'' to mean ''definable without parameters'', while other authors mean ''definable with parameters''.  Broadly speaking, the convention that ''definable'' means ''definable without parameters'' is more common amongst set theorists, while the opposite convention is more common amongst model theorists.
 
====Implicit definability====
Recall from above that an ''n''-ary relation ''R'' on the universe ''M'' of a structure <math>\mathcal{M}</math> is explicitly definable if there is a formula φ(''x''<sub>1</sub>,...,''x''<sub>''n''</sub>) such that
 
:<math>R = \{ (a_1,\ldots,a_n ) \in M^n : \mathcal{M} \vDash \phi(a_1,\ldots,a_n) \}</math>
 
Here the formula φ used to define a relation ''R'' must be over the signature of <math>\mathcal{M}</math> and so φ may not mention ''R'' itself, since ''R'' is not in the signature of <math>\mathcal{M}</math>.  If there is a formula φ in the extended language containing the language of <math>\mathcal{M}</math> and a new symbol ''R'', and the relation ''R'' is the only relation on <math>\mathcal{M}</math> such that <math>\mathcal{M} \vDash \phi</math>, then ''R'' is said to be '''implicitly definable''' over <math>\mathcal{M}</math>.
 
By Beth's theorem, every implicitly definable relation is explicitly definable.
 
==Many-sorted structures==
Structures as defined above are sometimes called '''{{visible anchor|one-sorted structure}}s''' to distinguish them from the more general '''{{visible anchor|many-sorted structure}}s'''. A many-sorted structure can have an arbitrary number of domains. The '''sorts''' are part of the signature, and they play the role of names for the different domains. [[Signature (logic)#Many-sorted signatures|Many-sorted signatures]] also prescribe on which sorts the functions and relations of a many-sorted structure are defined. Therefore the arities of function symbols or relation symbols must be more complicated objects such as tuples of sorts rather than natural numbers.
 
[[Vector space]]s, for example, can be regarded as two-sorted structures in the following way. The two-sorted signature of vector spaces consists of two sorts ''V'' (for vectors) and ''S'' (for scalars) and the following function symbols:
 
{| style="width:95%"
|-valign="top"
|
* +<sub>''S''</sub> and ×<sub>''S''</sub> of arity (''S'',&nbsp;''S'';&nbsp;''S'').
* −<sub>''S''</sub> of arity (''S'';&nbsp;''S'').
* 0<sub>''S''</sub> and 1<sub>''S''</sub> of arity (''S'').
|
* +<sub>''V''</sub> of arity (''V'',&nbsp;''V'';&nbsp;''V'').
* −<sub>''V''</sub> of arity (''V'';&nbsp;''V'').
* 0<sub>''V''</sub> of arity (''V'').
|
* × of arity (''S'',&nbsp;''V'';&nbsp;''V'').
|}
 
If ''V'' is a vector space over a field ''F'', the corresponding two-sorted structure <math>\mathcal V</math> consists of the vector domain <math>|\mathcal V|_V=V</math>, the scalar domain <math>|\mathcal V|_S=F</math>, and the obvious functions, such as the vector zero <math>0_V^{\mathcal V}=0\in|\mathcal V|_V</math>, the scalar zero <math>0_S^{\mathcal V}=0\in|\mathcal V|_S</math>, or scalar multiplication <math>\times^{\mathcal V}:|\mathcal V|_S\times|\mathcal V|_V\rightarrow|\mathcal V|_V</math>.
 
Many-sorted structures are often used as a convenient tool even when they could be avoided with a little effort. But they are rarely defined in a rigorous way, because it is straightforward and tedious (hence unrewarding) to carry out the generalization explicitly.
 
In most mathematical endeavours, not much attention is paid to the sorts. A [[many-sorted logic]] however naturally leads to a [[type theory]]. As [[Bart Jacobs]] puts it: "A logic is always a logic over a type theory." This emphasis in turn leads to [[categorical logic]] because a logic over a type theory categorically corresponds to one ("total") category, capturing the logic, being [[fibred category|fibred]] over another ("base") category, capturing the type theory.<ref>{{Citation
  | first = Bart
  | last = Jacobs
  | title = Categorical Logic and Type Theory
  | date = 1999
  | publisher = Elsevier | pages = 1–4}}</ref>
 
==Other generalizations==
===Partial algebras===
Both universal algebra and model theory study classes of (structures or) algebras that are defined by a signature and a set of axioms. In the case of model theory these axioms have the form of first-order sentences. The formalism of universal algebra is much more restrictive; essentially it only allows first-order sentences that have the form of universally quantified equations between terms, e.g. {{all}}&nbsp;''x''&nbsp;{{all}}''y''&nbsp;(''x''&nbsp;+&nbsp;''y''&nbsp;=&nbsp;''y''&nbsp;+&nbsp;''x''). One consequence is that the choice of a signature is more significant in universal algebra than it is in model theory. For example the class of groups, in the signature consisting of the binary function symbol × and the constant symbol 1, is an [[elementary class]], but it is not a [[Variety (universal algebra)|variety]]. Universal algebra solves this problem by adding a unary function symbol <sup>−1</sup>.
 
In the case of fields this strategy works only for addition. For multiplication it fails because 0 does not have a multiplicative inverse. An ad hoc attempt to deal with this would be to define 0<sup>−1</sup>&nbsp;=&nbsp;0. (This attempt fails, essentially because with this definition 0&nbsp;×&nbsp;0<sup>−1</sup>&nbsp;=&nbsp;1 is not true.) Therefore one is naturally led to allow partial functions, i.e., functions which are defined only on a subset of their domain. However, there are several obvious ways to generalize notions such as substructure, homomorphism and identity.
 
=== Structures for typed languages ===
In [[type theory]], there are many sorts of variables, each of which has a '''type'''. Types are inductively defined; given two types δ and σ there is also a type σ → δ that represents functions from objects of type σ to objects of type δ. A structure for a typed language (in the ordinary first-order semantics) must include a separate set of objects of each type, and for a function type the structure must have complete information about the function represented by each object of that type.
 
===Higher-order languages===
{{main|Second-order logic}}
There is more than one possible semantics for [[higher-order logic]], as discussed in the article on [[second-order logic]].  When using full higher-order semantics, a structure need only have a universe for objects of type 0, and the T-schema is extended so that a quantifier over a higher-order type is satisfied by the model if and only if it is disquotationally true.  When using first-order semantics, an additional sort is added for each higher-order type, as in the case of a many sorted first order language.
 
===Structures that are proper classes===
In the study of [[set theory]] and [[category theory]], it is sometimes useful to consider structures in which the domain of discourse is a [[proper class]] instead of a set. These structures are sometimes called '''class models''' to distinguish them from the "set models" discussed above. When the domain is a proper class, each function and relation symbol may also be represented by a proper class.
 
In [[Bertrand Russell]]'s [[Principia Mathematica]], structures were also allowed to have a proper class as their domain.
 
==Notes==
<references/>
 
== References ==
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* {{Citation | last1=Marker | first1=David | title=Model Theory: An Introduction | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-98760-6 | year=2002}}
* {{Citation | last1=Poizat | first1=Bruno | title=A Course in Model Theory: An Introduction to Contemporary Mathematical Logic | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-98655-5 | year=2000}}
* {{Citation |last=Rautenberg | first=Wolfgang | authorlink=Wolfgang Rautenberg | doi=10.1007/978-1-4419-1221-3 | title=A Concise Introduction to Mathematical Logic | url=http://www.springerlink.com/content/978-1-4419-1220-6/ | publisher=[[Springer Science+Business Media]] | location=[[New York City|New York]] | edition=3rd | isbn=978-1-4419-1220-6 | year=2010}}
* {{Citation | last1=Rothmaler | first1=Philipp | title=Introduction to Model Theory | publisher=[[CRC Press]] | location=London | isbn=978-90-5699-313-9 | year=2000}}
 
==External links==
*[http://plato.stanford.edu/entries/logic-classical/#4 Semantics] section in [http://plato.stanford.edu/entries/logic-classical/ Classical Logic] (an entry of [http://plato.stanford.edu Stanford Encyclopedia of Philosophy])
 
{{logic}}
 
[[Category:Mathematical structures]]
[[Category:Model theory]]
[[Category:Universal algebra]]
[[Category:Mathematical logic]]

Revision as of 05:18, 27 January 2014

Template:More footnotes In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations, and relations that are defined on it.

Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces. The term universal algebra is used for structures with no relation symbols.[1]

Model theory has a different scope that encompasses more arbitrary theories, including foundational structures such as models of set theory. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic. For a given theory in model theory, a structure is called a model, if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a semantic model when one discusses the notion in the more general setting of mathematical models. Logicians sometimes refer to structures as interpretations.[2]

In database theory, structures with no functions are studied as models for relational databases, in the form of relational models.

Definition

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Domain

The domain of a structure is an arbitrary set; it is also called the underlying set of the structure, its carrier (especially in universal algebra), or its universe (especially in model theory). In classical first-order logic, the definition of a structure prohibits the empty domain.[3]

Sometimes the notation dom⁡(𝒜) or |𝒜| is used for the domain of 𝒜, but often no notational distinction is made between a structure and its domain. (I.e. the same symbol 𝒜 refers both to the structure and its domain.)[4]

Signature

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. The signature of a structure consists of a set of function symbols and relation symbols along with a function that ascribes to each symbol s a natural number n=ar⁡(s) which is called the arity of s because it is the arity of the interpretation of s.

Since the signatures that arise in algebra often contain only function symbols, a signature with no relation symbols is called an algebraic signature. A structure with such a signature is also called an algebra; this should not be confused with the notion of an algebra over a field.

Interpretation function

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. The interpretation function I of 𝒜 assigns functions and relations to the symbols of the signature. Each function symbol f of arity n is assigned an n-ary function f𝒜=I(f) on the domain. Each relation symbol R of arity n is assigned an n-ary relation R𝒜=I(R)⊆Aar(R) on the domain. A nullary function symbol c is called a constant symbol, because its interpretation I(c) can be identified with a constant element of the domain.

When a structure (and hence an interpretation function) is given by context, no notational distinction is made between a symbol s and its interpretation I(s). For example if f is a binary function symbol of 𝒜, one simply writes f:𝒜2→𝒜 rather than f𝒜:|𝒜|2→|𝒜|.

Examples

The standard signature σf for fields consists of two binary function symbols + and ×, a unary function symbol −, and the two constant symbols 0 and 1. Thus a structure (algebra) for this signature consists of a set of elements A together with two binary functions, a unary function, and two distinguished elements; but there is no requirement that it satisfy any of the field axioms. The rational numbers Q, the real numbers R and the complex numbers C, like any other field, can be regarded as σ-structures in an obvious way:

𝒬=(Q,σf,I𝒬)
ℛ=(R,σf,Iℛ)
𝒞=(C,σf,I𝒞)

where

I𝒬(+):Q×Q→Q is addition of rational numbers,
I𝒬(×):Q×Q→Q is multiplication of rational numbers,
I𝒬(−):Q→Q is the function that takes each rational number x to -x, and
I𝒬(0)∈Q is the number 0 and
I𝒬(1)∈Q is the number 1;

and Iℛ and I𝒞 are similarly defined.

But the ring Z of integers, which is not a field, is also a σf-structure in the same way. In fact, there is no requirement that any of the field axioms hold in a σf-structure.

A signature for ordered fields needs an additional binary relation such as < or ≤, and therefore structures for such a signature are not algebras, even though they are of course algebraic structures in the usual, loose sense of the word.

The ordinary signature for set theory includes a single binary relation ∈. A structure for this signature consists of a set of elements and an interpretation of the ∈ relation as a binary relation on these elements.

Induced substructures and closed subsets

𝒜 is called an (induced) substructure of ℬ if

The usual notation for this relation is 𝒜⊆ℬ.

A subset B⊆|𝒜| of the domain of a structure 𝒜 is called closed if it is closed under the functions of 𝒜, i.e. if the following condition is satisfied: for every natural number n, every n-ary function symbol f (in the signature of 𝒜) and all elements b1,b2,…,bn∈B, the result of applying f to the n-tuple b1b2…bn is again an element of B: f(b1,b2,…,bn)∈B.

For every subset B⊆|𝒜| there is a smallest closed subset of |𝒜| that contains B. It is called the closed subset generated by B, or the hull of B, and denoted by ⟨B⟩ or ⟨B⟩𝒜. The operator ⟨⟩ is a finitary closure operator on the set of subsets of |𝒜|.

If 𝒜=(A,σ,I) and B⊆A is a closed subset, then (B,σ,I′) is an induced substructure of 𝒜, where I′ assigns to every symbol of σ the restriction to B of its interpretation in 𝒜. Conversely, the domain of an induced substructure is a closed subset.

The closed subsets (or induced substructures) of a structure form a lattice. The meet of two subsets is their intersection. The join of two subsets is the closed subset generated by their union. Universal algebra studies the lattice of substructures of a structure in detail.

Examples

Let σ = {+, ×, −, 0, 1} be again the standard signature for fields. When regarded as σ-structures in the natural way, the rational numbers form a substructure of the real numbers, and the real numbers form a substructure of the complex numbers. The rational numbers are the smallest substructure of the real (or complex) numbers that also satisfies the field axioms.

The set of integers gives an even smaller substructure of the real numbers which is not a field. Indeed, the integers are the substructure of the real numbers generated by the empty set, using this signature. The notion in abstract algebra that corresponds to a substructure of a field, in this signature, is that of a subring, rather than that of a subfield.

The most obvious way to define a graph is a structure with a signature σ consisting of a single binary relation symbol E. The vertices of the graph form the domain of the structure, and for two vertices a and b, (a,b)∈E  means that a and b are connected by an edge. In this encoding, the notion of induced substructure is more restrictive than the notion of subgraph. For example, let G be a graph consisting of two vertices connected by an edge, and let H be the graph consisting of the same vertices but no edges. H is a subgraph of G, but not an induced substructure. The notion in graph theory that corresponds to induced substructures is that of induced subgraphs.

Homomorphisms and embeddings

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Homomorphisms

Given two structures 𝒜 and ℬ of the same signature σ, a (σ-)homomorphism from 𝒜 to ℬ is a map h:|𝒜|→|ℬ| that preserves the functions and relations. More precisely:

  • For every n-ary function symbol f of σ and any elements a1,a2,…,an∈|𝒜|, the following equation holds:
h(f(a1,a2,…,an))=f(h(a1),h(a2),…,h(an)).
  • For every n-ary relation symbol R of σ and any elements a1,a2,…,an∈|𝒜|, the following implication holds:
(a1,a2,…,an)∈R⟹(h(a1),h(a2),…,h(an))∈R.

The notation for a homomorphism h from 𝒜 to ℬ is h:𝒜→ℬ.

For every signature σ there is a concrete category σ-Hom which has σ-structures as objects and σ-homomorphisms as morphisms.

A homomorphism h:𝒜→ℬ is sometimes called strong if for every n-ary relation symbol R and any elements b1,b2,…,bn∈|ℬ| such that (b1,b2,…,bn)∈R, there are a1,a2,…,an∈|𝒜| such that (a1,a2,…,an)∈R and b1=h(a1),b2=h(a2),…,bn=h(an). The strong homomorphisms give rise to a subcategory of σ-Hom.

Embeddings

A (σ-)homomorphism h:𝒜→ℬ is called a (σ-)embedding if it is one-to-one and

  • for every n-ary relation symbol R of σ and any elements a1,a2,…,an, the following equivalence holds:
(a1,a2,…,an)∈R⟺(h(a1),h(a2),…,h(an))∈R.

Thus an embedding is the same thing as a strong homomorphism which is one-to-one. The category σ-Emb of σ-structures and σ-embeddings is a concrete subcategory of σ-Hom.

Induced substructures correspond to subobjects in σ-Emb. If σ has only function symbols, σ-Emb is the subcategory of monomorphisms of σ-Hom. In this case induced substructures also correspond to subobjects in σ-Hom.

Example

As seen above, in the standard encoding of graphs as structures the induced substructures are precisely the induced subgraphs. However, a homomorphism between graphs is the same thing as a homomorphism between the two structures coding the graph. In the example of the previous section, even though the subgraph H of G is not induced, the identity map id: H → G is a homomorphism. This map is in fact a monomorphism in the category σ-Hom, and therefore H is a subobject of G which is not an induced substructure.

Homomorphism problem

The following problem is known as the homomorphism problem:

Given two finite structures 𝒜 and ℬ of a finite relational signature, find a homomorphism h:𝒜→ℬ or show that no such homomorphism exists.

Every constraint satisfaction problem (CSP) has a translation into the homomorphism problem.[5] Therefore the complexity of CSP can be studied using the methods of finite model theory.

Another application is in database theory, where a relational model of a database is essentially the same thing as a relational structure. It turns out that a conjunctive query on a database can be described by another structure in the same signature as the database model. A homomorphism from the relational model to the structure representing the query is the same thing as a solution to the query. This shows that the conjunctive query problem is also equivalent to the homomorphism problem.

Structures and first-order logic

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In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. Structures are sometimes referred to as "first-order structures". This is misleading, as nothing in their definition ties them to any specific logic, and in fact they are suitable as semantic objects both for very restricted fragments of first-order logic such as that used in universal algebra, and for second-order logic. In connection with first-order logic and model theory, structures are often called models, even when the question "models of what?" has no obvious answer.

Satisfaction relation

Each first-order structure ℳ has a satisfaction relation ℳ⊨ϕ defined for all formulas ϕ in the language consisting of the language of ℳ together with a constant symbol for each element of M, which is interpreted as that element. This relation is defined inductively using Tarski's T-schema.

A structure ℳ is said to be a model of a theory T if the language of ℳ is the same as the language of T and every sentence in T is satisfied by ℳ. Thus, for example, a "ring" is a structure for the language of rings that satisfies each of the ring axioms, and a model of ZFC set theory is a structure in the language of set theory that satisfies each of the ZFC axioms.

Definable relations

An n-ary relation R on the universe M of a structure ℳ is said to be definable (or explicitly definable, or ∅-definable) if there is a formula φ(x1,...,xn) such that

R={(a1,…,an)∈Mn:ℳ⊨ϕ(a1,…,an)}.

In other words, R is definable if and only if there is a formula φ such that

(a1,…,an)∈R⇔ℳ⊨ϕ(a1,…,an)

is correct.

An important special case is the definability of specific elements. An element m of M is definable in ℳ if and only if there is a formula φ(x) such that

ℳ⊨∀x(x=m↔ϕ(x)).

Definability with parameters

A relation R is said to be definable with parameters (or |ℳ|-definable) if there is a formula φ with parameters from ℳ such that R is definable using φ. Every element of a structure is definable using the element itself as a parameter.

It should be noted that some authors use definable to mean definable without parameters, while other authors mean definable with parameters. Broadly speaking, the convention that definable means definable without parameters is more common amongst set theorists, while the opposite convention is more common amongst model theorists.

Implicit definability

Recall from above that an n-ary relation R on the universe M of a structure ℳ is explicitly definable if there is a formula φ(x1,...,xn) such that

R={(a1,…,an)∈Mn:ℳ⊨ϕ(a1,…,an)}

Here the formula φ used to define a relation R must be over the signature of ℳ and so φ may not mention R itself, since R is not in the signature of ℳ. If there is a formula φ in the extended language containing the language of ℳ and a new symbol R, and the relation R is the only relation on ℳ such that ℳ⊨ϕ, then R is said to be implicitly definable over ℳ.

By Beth's theorem, every implicitly definable relation is explicitly definable.

Many-sorted structures

Structures as defined above are sometimes called Template:Visible anchors to distinguish them from the more general Template:Visible anchors. A many-sorted structure can have an arbitrary number of domains. The sorts are part of the signature, and they play the role of names for the different domains. Many-sorted signatures also prescribe on which sorts the functions and relations of a many-sorted structure are defined. Therefore the arities of function symbols or relation symbols must be more complicated objects such as tuples of sorts rather than natural numbers.

Vector spaces, for example, can be regarded as two-sorted structures in the following way. The two-sorted signature of vector spaces consists of two sorts V (for vectors) and S (for scalars) and the following function symbols:

  • +S and ×S of arity (S, S; S).
  • −S of arity (S; S).
  • 0S and 1S of arity (S).
  • +V of arity (V, V; V).
  • −V of arity (V; V).
  • 0V of arity (V).
  • × of arity (S, V; V).

If V is a vector space over a field F, the corresponding two-sorted structure 𝒱 consists of the vector domain |𝒱|V=V, the scalar domain |𝒱|S=F, and the obvious functions, such as the vector zero 0V𝒱=0∈|𝒱|V, the scalar zero 0S𝒱=0∈|𝒱|S, or scalar multiplication ×𝒱:|𝒱|S×|𝒱|V→|𝒱|V.

Many-sorted structures are often used as a convenient tool even when they could be avoided with a little effort. But they are rarely defined in a rigorous way, because it is straightforward and tedious (hence unrewarding) to carry out the generalization explicitly.

In most mathematical endeavours, not much attention is paid to the sorts. A many-sorted logic however naturally leads to a type theory. As Bart Jacobs puts it: "A logic is always a logic over a type theory." This emphasis in turn leads to categorical logic because a logic over a type theory categorically corresponds to one ("total") category, capturing the logic, being fibred over another ("base") category, capturing the type theory.[6]

Other generalizations

Partial algebras

Both universal algebra and model theory study classes of (structures or) algebras that are defined by a signature and a set of axioms. In the case of model theory these axioms have the form of first-order sentences. The formalism of universal algebra is much more restrictive; essentially it only allows first-order sentences that have the form of universally quantified equations between terms, e.g. In computational number theory and computational algebra, Pollard's kangaroo algorithm (aka Pollard's lambda algorithm, see Naming below) is an algorithm for solving the discrete logarithm problem. The algorithm was introduced in 1978 by the number theorist J. M. Pollard, in the same paper [7] as his better-known rho algorithm for solving the same problem. Although Pollard described the application of his algorithm to the discrete logarithm problem in the multiplicative group of units modulo a prime p, it is in fact a generic discrete logarithm algorithm—it will work in any finite cyclic group.

The algorithm

Suppose G is a finite cyclic group of order n which is generated by the element α, and we seek to find the discrete logarithm x of the element β to the base α. In other words, we seek x∈Zn such that αx=β. The lambda algorithm allows us to search for x in some subset {a,…,b}⊂Zn. We may search the entire range of possible logarithms by setting a=0 and b=n−1, although in this case Pollard's rho algorithm is more efficient. We proceed as follows:

1. Choose a set S of integers and define a pseudorandom map f:G→S.

2. Choose an integer N and compute a sequence of group elements {x0,x1,…,xN} according to:

3. Compute

d=∑i=0N−1f(xi).

Observe that:

xN=x0αd=αb+d.

4. Begin computing a second sequence of group elements {y0,y1,…} according to:

and a corresponding sequence of integers {d0,d1,…} according to:

dn=∑i=0n−1f(yi).

Observe that:

yi=y0αdi=βαdi for i=0,1,…,N−1

5. Stop computing terms of {yi} and {di} when either of the following conditions are met:

A) yj=xN for some j. If the sequences {xi} and {yj} "collide" in this manner, then we have:
xN=yj⇒αb+d=βαdj⇒β=αb+d−dj(modn)⇒x≡b+d−dj(modn)
and so we are done.
B) di>b−a+d. If this occurs, then the algorithm has failed to find x. Subsequent attempts can be made by changing the choice of S and/or f.

Complexity

Pollard gives the time complexity of the algorithm as O(b−a), based on a probabilistic argument which follows from the assumption that f acts pseudorandomly. Note that when the size of the set {a, …, b} to be searched is measured in bits, as is normal in complexity theory, the set has size log(b − a), and so the algorithm's complexity is O(b−a)=O(212log⁡(b−a)), which is exponential in the problem size. For this reason, Pollard's lambda algorithm is considered an exponential time algorithm. For an example of a subexponential time discrete logarithm algorithm, see the index calculus algorithm.

Naming

The algorithm is well known by two names.

The first is "Pollard's kangaroo algorithm". This name is a reference to an analogy used in the paper presenting the algorithm, where the algorithm is explained in terms of using a tame kangaroo to trap a wild kangaroo. Pollard has explained[8] that this analogy was inspired by a "fascinating" article published in the same issue of Scientific American as an exposition of the RSA public key cryptosystem. The article[9] described an experiment in which a kangaroo's "energetic cost of locomotion, measured in terms of oxygen consumption at various speeds, was determined by placing kangaroos on a treadmill".

The second is "Pollard's lambda algorithm". Much like the name of another of Pollard's discrete logarithm algorithms, Pollard's rho algorithm, this name refers to the similarity between a visualisation of the algorithm and the Greek letter lambda (λ). The shorter stroke of the letter lambda corresponds to the sequence {xi}, since it starts from the position b to the right of x. Accordingly, the longer stroke corresponds to the sequence {yi}, which "collides with" the first sequence (just like the strokes of a lambda intersect) and then follows it subsequently.

Pollard has expressed a preference for the name "kangaroo algorithm",[10] as this avoids confusion with some parallel versions of his rho algorithm, which have also been called "lambda algorithms".

See also

References

  1. ↑ Some authors refer to structures as "algebras" when generalizing universal algebra to allow relations as well as functions.
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  3. ↑ This is similar to the definition of a prime number in elementary number theory, which has been carefully chosen so that the irreducible number 1 is not considered prime. The convention that the domain of a structure may not be empty is particularly important in logic, because several common inference rules, notably, universal instantiation, are not sound when empty structures are permitted. A logical system that allows the empty domain is known as an inclusive logic.
  4. ↑ As a consequence of these conventions, the notation |𝒜| may also be used to refer to the cardinality of the domain of 𝒜. In practice this never leads to confusion.
  5. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  6. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  7. ↑ J. Pollard, Monte Carlo methods for index computation mod p, Mathematics of Computation, Volume 32, 1978
  8. ↑ J. M. Pollard, Kangaroos, Monopoly and Discrete Logarithms, Journal of Cryptology, Volume 13, pp 437-447, 2000
  9. ↑ T. J. Dawson, Kangaroos, Scientific American, August 1977, pp. 78-89
  10. ↑ http://sites.google.com/site/jmptidcott2/

Template:Number-theoretic algorithms x In computational number theory and computational algebra, Pollard's kangaroo algorithm (aka Pollard's lambda algorithm, see Naming below) is an algorithm for solving the discrete logarithm problem. The algorithm was introduced in 1978 by the number theorist J. M. Pollard, in the same paper [1] as his better-known rho algorithm for solving the same problem. Although Pollard described the application of his algorithm to the discrete logarithm problem in the multiplicative group of units modulo a prime p, it is in fact a generic discrete logarithm algorithm—it will work in any finite cyclic group.

The algorithm

Suppose G is a finite cyclic group of order n which is generated by the element α, and we seek to find the discrete logarithm x of the element β to the base α. In other words, we seek x∈Zn such that αx=β. The lambda algorithm allows us to search for x in some subset {a,…,b}⊂Zn. We may search the entire range of possible logarithms by setting a=0 and b=n−1, although in this case Pollard's rho algorithm is more efficient. We proceed as follows:

1. Choose a set S of integers and define a pseudorandom map f:G→S.

2. Choose an integer N and compute a sequence of group elements {x0,x1,…,xN} according to:

3. Compute

d=∑i=0N−1f(xi).

Observe that:

xN=x0αd=αb+d.

4. Begin computing a second sequence of group elements {y0,y1,…} according to:

and a corresponding sequence of integers {d0,d1,…} according to:

dn=∑i=0n−1f(yi).

Observe that:

yi=y0αdi=βαdi for i=0,1,…,N−1

5. Stop computing terms of {yi} and {di} when either of the following conditions are met:

A) yj=xN for some j. If the sequences {xi} and {yj} "collide" in this manner, then we have:
xN=yj⇒αb+d=βαdj⇒β=αb+d−dj(modn)⇒x≡b+d−dj(modn)
and so we are done.
B) di>b−a+d. If this occurs, then the algorithm has failed to find x. Subsequent attempts can be made by changing the choice of S and/or f.

Complexity

Pollard gives the time complexity of the algorithm as O(b−a), based on a probabilistic argument which follows from the assumption that f acts pseudorandomly. Note that when the size of the set {a, …, b} to be searched is measured in bits, as is normal in complexity theory, the set has size log(b − a), and so the algorithm's complexity is O(b−a)=O(212log⁡(b−a)), which is exponential in the problem size. For this reason, Pollard's lambda algorithm is considered an exponential time algorithm. For an example of a subexponential time discrete logarithm algorithm, see the index calculus algorithm.

Naming

The algorithm is well known by two names.

The first is "Pollard's kangaroo algorithm". This name is a reference to an analogy used in the paper presenting the algorithm, where the algorithm is explained in terms of using a tame kangaroo to trap a wild kangaroo. Pollard has explained[2] that this analogy was inspired by a "fascinating" article published in the same issue of Scientific American as an exposition of the RSA public key cryptosystem. The article[3] described an experiment in which a kangaroo's "energetic cost of locomotion, measured in terms of oxygen consumption at various speeds, was determined by placing kangaroos on a treadmill".

The second is "Pollard's lambda algorithm". Much like the name of another of Pollard's discrete logarithm algorithms, Pollard's rho algorithm, this name refers to the similarity between a visualisation of the algorithm and the Greek letter lambda (λ). The shorter stroke of the letter lambda corresponds to the sequence {xi}, since it starts from the position b to the right of x. Accordingly, the longer stroke corresponds to the sequence {yi}, which "collides with" the first sequence (just like the strokes of a lambda intersect) and then follows it subsequently.

Pollard has expressed a preference for the name "kangaroo algorithm",[4] as this avoids confusion with some parallel versions of his rho algorithm, which have also been called "lambda algorithms".

See also

References

  1. ↑ J. Pollard, Monte Carlo methods for index computation mod p, Mathematics of Computation, Volume 32, 1978
  2. ↑ J. M. Pollard, Kangaroos, Monopoly and Discrete Logarithms, Journal of Cryptology, Volume 13, pp 437-447, 2000
  3. ↑ T. J. Dawson, Kangaroos, Scientific American, August 1977, pp. 78-89
  4. ↑ http://sites.google.com/site/jmptidcott2/

Template:Number-theoretic algorithmsy (x + y = y + x). One consequence is that the choice of a signature is more significant in universal algebra than it is in model theory. For example the class of groups, in the signature consisting of the binary function symbol × and the constant symbol 1, is an elementary class, but it is not a variety. Universal algebra solves this problem by adding a unary function symbol −1.

In the case of fields this strategy works only for addition. For multiplication it fails because 0 does not have a multiplicative inverse. An ad hoc attempt to deal with this would be to define 0−1 = 0. (This attempt fails, essentially because with this definition 0 × 0−1 = 1 is not true.) Therefore one is naturally led to allow partial functions, i.e., functions which are defined only on a subset of their domain. However, there are several obvious ways to generalize notions such as substructure, homomorphism and identity.

Structures for typed languages

In type theory, there are many sorts of variables, each of which has a type. Types are inductively defined; given two types δ and σ there is also a type σ → δ that represents functions from objects of type σ to objects of type δ. A structure for a typed language (in the ordinary first-order semantics) must include a separate set of objects of each type, and for a function type the structure must have complete information about the function represented by each object of that type.

Higher-order languages

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. There is more than one possible semantics for higher-order logic, as discussed in the article on second-order logic. When using full higher-order semantics, a structure need only have a universe for objects of type 0, and the T-schema is extended so that a quantifier over a higher-order type is satisfied by the model if and only if it is disquotationally true. When using first-order semantics, an additional sort is added for each higher-order type, as in the case of a many sorted first order language.

Structures that are proper classes

In the study of set theory and category theory, it is sometimes useful to consider structures in which the domain of discourse is a proper class instead of a set. These structures are sometimes called class models to distinguish them from the "set models" discussed above. When the domain is a proper class, each function and relation symbol may also be represented by a proper class.

In Bertrand Russell's Principia Mathematica, structures were also allowed to have a proper class as their domain.

Notes


References

  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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