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In [[order theory|mathematical order theory]], an '''order-embedding''' is a special kind of [[monotone function]], which provides a way to include one [[partially ordered set]] into another. Like [[Galois connection]]s, order-embeddings constitute a notion which is strictly weaker than the concept of an [[order isomorphism]]. Both of these weakenings may be understood in terms of [[category theory]]. | |||
== Formal definition == | |||
Formally, given two partially ordered sets (''S'', ≤) and (''T'', ≤), a function ''f'': ''S'' → ''T'' is an ''order-embedding'' if ''f'' is both [[order-preserving]] and [[order-reflecting]], i.e. for all ''x'' and ''y'' in ''S'', one has | |||
: <math>x\leq y \text{ if and only if } f(x)\leq f(y).</math><ref name="dp02">{{citation | |||
| last1 = Davey | first1 = B. A. | |||
| last2 = Priestley | first2 = H. A. | |||
| contribution = Maps between ordered sets | |||
| edition = 2nd | |||
| isbn = 0-521-78451-4 | |||
| location = New York | |||
| mr = 1902334 | |||
| pages = 23–24 | |||
| publisher = Cambridge University Press | |||
| title = Introduction to Lattices and Order | |||
| url = http://books.google.com/books?id=vVVTxeuiyvQC&pg=PA23 | |||
| year = 2002}}.</ref> | |||
Note that such a function is necessarily [[injective]], since ''f''(''x'') = ''f''(''y'') implies ''x'' ≤ ''y'' and ''y'' ≤ ''x''.<ref name="dp02"/> If an order-embedding between two posets ''S'' and ''T'' exists, one says that ''S'' can be embedded into ''T''. | |||
== Properties == | |||
An order isomorphism can be characterized as a [[surjective]] order-embedding. As a consequence, any order-embedding ''f'' restricts to an isomorphism between its [[domain (mathematics)|domain]] ''S'' and its [[range (mathematics)|range]] ''f''(''S''), which justifies the term "embedding".<ref name="dp02"/> On the other hand, it might well be that two (necessarily infinite) posets are mutually embeddable into each other without being isomorphic. An example is provided by the set of [[real number]]s and its [[interval (mathematics)|interval]] [−1,1]. Ordering both sets in the natural way, one clearly finds that [−1,1] can be embedded into the reals. On the other hand, one can define a function ''e'' from the real numbers to [−1,1] as<ref>For a very similar example using arctan to order-embed the real numbers into an interval, see {{citation|title=Discovering Modern Set Theory: The basics|volume=8|series=Fields Institute Monographs|first1=Winfried|last1=Just|first2=Martin|last2=Weese|publisher=American Mathematical Society|year=1996|isbn=9780821872475|page=21|url=http://books.google.com/books?id=TPvHr7fcvHoC&pg=PA21}}.</ref> | |||
: <math>e(x) = \frac{2}{\pi}\arctan x</math> | |||
This is the projection of the real number line to (half of) the circle with circumference 4 (see [[trigonometric function]]s for details) and embeds the reals into [−1,1]. Yet, the two posets are not isomorphic: [−1,1] has both a [[least element|least]] and a [[greatest element]], which are not present in the case of the real numbers. This shows that an isomorphism cannot exist. | |||
In a retract (a pair of order-preserving maps whose composition is the identity), the first of the two maps (called a coretraction) must be an order-embedding.<ref>{{citation | |||
| last1 = Duffus | first1 = Dwight | |||
| last2 = Laflamme | first2 = Claude | |||
| last3 = Pouzet | first3 = Maurice | |||
| arxiv = math/0612458 | |||
| doi = 10.1007/s00012-008-2125-6 | |||
| issue = 1-2 | |||
| journal = Algebra Universalis | |||
| mr = 2453498 | |||
| pages = 243–255 | |||
| title = Retracts of posets: the chain-gap property and the selection property are independent | |||
| volume = 59 | |||
| year = 2008}}.</ref> However, not every order-embedding is a coretraction. As a trivial example of this phenomenon, the unique order embedding from the empty poset to a nonempty poset ''P'' has no retract, because there is no order-preserving map from ''P'' to the empty poset. More illustratively, consider the "diamond poset" with elements {00, 01, 10, 11} with 00 < 01 < 11, 00 < 10 < 11 and 01 incomparable to 10. Consider the embedded sub-poset "V" consisting of {00, 01, 10} (with 00 < 01 and 00 < 10 and 01 incomparable to 10). A retract of the embedding V -> diamond would need to send 11 to somewhere in "V" above both 01 and 10, but there is no such place. | |||
== Additional Perspectives == | |||
{{unreferenced section|date=October 2013}} | |||
Posets can straightforwardly be viewed from many perspectives, and order embeddings are basic enough that they tend to be visible from everywhere. For example: | |||
* (Model theoretically) A poset is a set equipped with a (reflexive, antisymmetric, transitive) binary relation. An order embedding A -> B is an isomorphism from A to an [[elementary substructure]] of B. | |||
* (Graph theoretically) A poset is a (transitive, acyclic, directed, reflexive) graph. An order embedding A -> B is a [[graph isomorphism]] from A to an [[induced subgraph]] of B. | |||
* (Category theoretically) A poset is a (small, skeletal) category such that each homset has at most one element. An order embedding A -> B is a full and faithful functor from A to B which is injective on objects, or equivalently an isomorphism from A to a [[full subcategory]] of B. | |||
==References== | |||
{{reflist}} | |||
[[Category:Order theory]] | |||
Revision as of 16:33, 3 February 2014
In mathematical order theory, an order-embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order-embeddings constitute a notion which is strictly weaker than the concept of an order isomorphism. Both of these weakenings may be understood in terms of category theory.
Formal definition
Formally, given two partially ordered sets (S, ≤) and (T, ≤), a function f: S → T is an order-embedding if f is both order-preserving and order-reflecting, i.e. for all x and y in S, one has
Note that such a function is necessarily injective, since f(x) = f(y) implies x ≤ y and y ≤ x.[1] If an order-embedding between two posets S and T exists, one says that S can be embedded into T.
Properties
An order isomorphism can be characterized as a surjective order-embedding. As a consequence, any order-embedding f restricts to an isomorphism between its domain S and its range f(S), which justifies the term "embedding".[1] On the other hand, it might well be that two (necessarily infinite) posets are mutually embeddable into each other without being isomorphic. An example is provided by the set of real numbers and its interval [−1,1]. Ordering both sets in the natural way, one clearly finds that [−1,1] can be embedded into the reals. On the other hand, one can define a function e from the real numbers to [−1,1] as[2]
This is the projection of the real number line to (half of) the circle with circumference 4 (see trigonometric functions for details) and embeds the reals into [−1,1]. Yet, the two posets are not isomorphic: [−1,1] has both a least and a greatest element, which are not present in the case of the real numbers. This shows that an isomorphism cannot exist.
In a retract (a pair of order-preserving maps whose composition is the identity), the first of the two maps (called a coretraction) must be an order-embedding.[3] However, not every order-embedding is a coretraction. As a trivial example of this phenomenon, the unique order embedding from the empty poset to a nonempty poset P has no retract, because there is no order-preserving map from P to the empty poset. More illustratively, consider the "diamond poset" with elements {00, 01, 10, 11} with 00 < 01 < 11, 00 < 10 < 11 and 01 incomparable to 10. Consider the embedded sub-poset "V" consisting of {00, 01, 10} (with 00 < 01 and 00 < 10 and 01 incomparable to 10). A retract of the embedding V -> diamond would need to send 11 to somewhere in "V" above both 01 and 10, but there is no such place.
Additional Perspectives
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Posets can straightforwardly be viewed from many perspectives, and order embeddings are basic enough that they tend to be visible from everywhere. For example:
- (Model theoretically) A poset is a set equipped with a (reflexive, antisymmetric, transitive) binary relation. An order embedding A -> B is an isomorphism from A to an elementary substructure of B.
- (Graph theoretically) A poset is a (transitive, acyclic, directed, reflexive) graph. An order embedding A -> B is a graph isomorphism from A to an induced subgraph of B.
- (Category theoretically) A poset is a (small, skeletal) category such that each homset has at most one element. An order embedding A -> B is a full and faithful functor from A to B which is injective on objects, or equivalently an isomorphism from A to a full subcategory of B.
References
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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. - ↑ For a very similar example using arctan to order-embed the real numbers into an interval, see 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.