Equivalent rectangular bandwidth

In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for any locally finite partially ordered set and commutative ring with unity.

Definition

A locally finite poset is one for which every closed interval

[a, b] = {x : a ≤ x ≤ b}

within it is finite.

The members of the incidence algebra are the functions f assigning to each nonempty interval [a, b] a scalar f(a, b), which is taken from the ring of scalars, a commutative ring with unity. On this underlying set one defines addition and scalar multiplication pointwise, and "multiplication" in the incidence algebra is a convolution defined by

$(f*g)(a,b)=\sum _{a\le x\le b}f(a,x)g(x,b).$

An incidence algebra is finite-dimensional if and only if the underlying poset is finite.

Related concepts

An incidence algebra is analogous to a group algebra; indeed, both the group algebra and the incidence algebra are special cases of a categorical algebra, defined analogously; groups and posets being special kinds of categories.

Special elements

The multiplicative identity element of the incidence algebra is the delta function, defined by

$\delta (a,b)=\{\begin{array}{ll}1& \text{if }a=b\\ 0& \text{if }a<b.\end{array}$

The zeta function of an incidence algebra is the constant function ζ(a, b) = 1 for every interval [a, b]. Multiplying by ζ is analogous to integration.

One can show that ζ is invertible in the incidence algebra (with respect to the convolution defined above). (Generally, a member h of the incidence algebra is invertible if and only if h(x, x) is invertible for every x.) The multiplicative inverse of the zeta function is the Möbius function μ(a, b); every value of μ(a, b) is an integral multiple of 1 in the base ring.

The Möbius function can also be defined directly, by the following relation:

$\mu (x,y)=\{\begin{array}{ll}1& \text{<mrow data-mjx-texclass="ORD">if</mrow>}x=y\\ -\sum _{z:x\le z<y}\mu (x,z)& \text{<mrow data-mjx-texclass="ORD">for</mrow>}x<y\\ 0& \text{<mrow data-mjx-texclass="ORD">otherwise</mrow>}.\end{array}$

Multiplying by μ is analogous to differentiation, and is called Möbius inversion.

Examples

• Positive integers ordered by divisibility

The Möbius function is μ(a, b) = μ(b/a), where the second "μ" is the classical Möbius function introduced into number theory in the 19th century.

• Finite subsets of some set E, ordered by inclusion

The Möbius function is

$\mu (S,T)=(-1{)}^{|T\setminus S|}$

whenever S and T are finite subsets of E with S ⊆ T, and Möbius inversion is called the principle of inclusion-exclusion.

Geometrically, this is a hypercube: $2^E=\{0,1{\}}^E.$

• Natural numbers with their usual order

The Möbius function is

$\mu (x,y)=\{\begin{array}{ll}1& \text{if }y-x=0,\\ -1& \text{if }y-x=1,\\ 0& \text{if }y-x>1,\end{array}$

and Möbius inversion is called the (backwards) difference operator.

Geometrically, this corresponds to the discrete number line.

Recall that convolution of sequences corresponds to multiplication of formal power series.

The Möbius function corresponds to the sequence (1, −1, 0, 0, 0, ... ) of coefficients of the formal power series 1 − z, and the zeta function in this case corresponds to the sequence of coefficients (1, 1, 1, 1, ... ) of the formal power series $(1-z{)}^{-1}=1+z+z^2+z^3+\cdots$, which is inverse. The delta function in this incidence algebra similarly corresponds to the formal power series 1.

• Subgroups of a finite p-group G, ordered by inclusion

The Möbius function is

${\mu }_G(H_1,H_2)=(-1{)}^k{p}^{(\genfrac{}{}{0ex}{}{k}{2})}$ if $H_1$ is a normal subgroup of $H_2$ and $H_2/H_1\cong (\mathbf{Z}/p\mathbf{Z}{)}^k$

and it is 0 otherwise. This is a theorem of Weisner (1935).

• Finite sub-multisets of some multiset E, ordered by inclusion

The above three examples can be unified and generalized by considering a multiset E, and finite sub-multisets S and T of E. The Möbius function is

$\mu (S,T)=\{\begin{array}{ll}0& \text{if }T\setminus S\text{ is a proper multiset (has repeated elements)}\\ (-1{)}^{|T\setminus S|}& \text{if }T\setminus S\text{ is a set (has no repeated elements)}.\end{array}$

This generalizes the positive integers ordered by divisibility by a positive integer corresponding to its multiset of prime divisors with multiplicity, e.g., 12 corresponds to the multiset $\{2,2,3\}.$

This generalizes the natural numbers with their usual order by a natural number corresponding to a multiset of one underlying element and cardinality equal to that number, e.g., 3 corresponds to the multiset $\{1,1,1\}.$

• Partitions of a set

Partially order the set of all partitions of a finite set by saying σ ≤ τ if σ is a finer partition than τ. Then the Möbius function is

$\mu (\sigma ,\tau )=(-1{)}^{n-r}(2!{)}^{r_3}(3!{)}^{r_4}\cdots ((n-1)!{)}^{r_n}$

where n is the number of blocks in the finer partition σ, r is the number of blocks in the coarser partition τ, and r_i is the number of blocks of τ that contain exactly i blocks of σ.

Euler characteristic

A poset is bounded if it has smallest and largest elements, which we call 0 and 1 respectively (not to be confused with the 0 and 1 of the ring of scalars). The Euler characteristic of a bounded finite poset is μ(0,1). The reason for this terminology is the following: If P has a 0 and 1, then μ(0,1) is the reduced Euler characteristic of the simplicial complex whose faces are chains in P\{0, 1}.

Reduced incidence algebras

Any member of an incidence algebra that assigns the same value to any two intervals that are isomorphic to each other as posets is a member of the reduced incidence algebra. This is a subalgebra of the incidence algebra, and it clearly contains the incidence algebra's identity element and zeta function. Any element of the reduced incidence algebra that is invertible in the larger incidence algebra has its inverse in the reduced incidence algebra. As a consequence, the Möbius function is always a member of the reduced incidence algebra. Reduced incidence algebras shed light on the theory of generating functions, as alluded to in the case of the natural numbers above.

See also

• Graph algebra
• Incidence coalgebra
• Path algebra

Literature

Incidence algebras of locally finite posets were treated in a number of papers of Gian-Carlo Rota beginning in 1964, and by many later combinatorialists. Rota's 1964 paper was:

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• N. Jacobson, Basic Algebra. I, W. H. Freeman and Co., 1974. See section 8.6 for a treatment of Mobius functions on posets

Further reading

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  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.
  
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