Incidence algebra

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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.

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[edit] Definition

A locally finite poset is one for which every closed interval

[a, b] = {x : a ≤ x ≤ b}

within it is finite.

The commutative ring with unity is called the ring of scalars.

The members of the incidence algebra are the functions f assigning to each interval [a, b] a scalar f(a, b). 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\leq x\leq b}f(a, x)g(x, b).

An incidence algebra is finite-dimensional if and only if the underlying partially ordered set is finite.

[edit] 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.

[edit] Special elements

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

\delta(a, b) = \left\{ \begin{matrix} \,1, & \mbox{if } a=b \\ \,0, & \mbox{if } a<b \end{matrix} \right.

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 field.

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

[edit] Examples

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)^{\left|T\setminus S\right|}
whenever S and T are finite subsets of E with S ⊆ T, and Möbius inversion is called the principle of inclusion-exclusion.
  • Natural numbers with their usual order
The Möbius function is
\mu(x,y)=\left\{\begin{matrix} 1 & \mbox{if }y-x=0, \\ -1 & \mbox{if }y-x=1, \\ 0 & \mbox{if }y-x>1. \end{matrix}\right.
and Möbius inversion is called the (backwards) difference operator.
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.
  • 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 ri is the number of blocks of τ that contain exactly i blocks of σ.

[edit] 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); it is always an integer. This concept is related to the classical Euler characteristic.

[edit] 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.

[edit] 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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