Incidence algebra


In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set
and commutative ring with unity. Subalgebras called reduced incidence algebras give a natural construction of various types of generating functions used in combinatorics and number theory.

Definition

A locally finite poset is one in which every closed interval
is finite.
The members of the incidence algebra are the functions f assigning to each nonempty interval a scalar f, 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
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 category 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
The zeta function of an incidence algebra is the constant function ζ = 1 for every nonempty interval . Multiplying by ζ is analogous to integration.
One can show that ζ is invertible in the incidence algebra. The multiplicative inverse of the zeta function is the Möbius function μ; every value of μ is an integral multiple of 1 in the base ring.
The Möbius function can also be defined inductively by the following relation:
Multiplying by μ is analogous to differentiation, and is called Möbius inversion.
The square of the zeta function counts the number of elements in an interval:

Examples

A poset is bounded if it has smallest and largest elements, which we call 0 and 1 respectively. The Euler characteristic of a bounded finite poset is μ. The reason for this terminology is the following: If P has a 0 and 1, then μ is the reduced Euler characteristic of the simplicial complex whose faces are chains in P \ . This can be shown using Philip Hall's theorem, relating the value of μ to the number of chains of length i.

Reduced incidence algebras

The reduced incidence algebra consists of functions which assign the same value to any two intervals which are equivalent in an appropriate sense, usually meaning isomorphic as posets. 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. Thus the Möbius function is also in the reduced incidence algebra.
Reduced incidence algebras were introduced by Doubillet, Rota, and Stanley to give a natural construction of various rings of generating functions.

Natural numbers and ordinary generating functions

For the poset the reduced incidence algebra consists of functions invariant under translation, for all so as to have the same value on isomorphic intervals and . Let t denote the function with t = 1 and t = 0 otherwise, a kind of invariant delta function on isomorphism classes of intervals. Its powers in the incidence algebra are the other invariant delta functions tn = 1 and tn = 0 otherwise. These form a basis for the reduced incidence algebra, and we may write any invariant function as. This notation makes clear the isomorphism between the reduced incidence algebra and the ring of formal power series over the scalars R, also known as the ring of ordinary generating functions. We may write the zeta function as the reciprocal of the Möbius function

Subset poset and exponential generating functions

For the Boolean poset of finite subsets ordered by inclusion, the reduced incidence algebra consists of invariant functions defined to have the same value on isomorphic intervals and with |T\S| = |T'\S'|. Again, let t denote the invariant delta function with t = 1 for |T\S| = 1 and t = 0 otherwise. Its powers are:
where the sum is over all chains and the only non-zero terms occur for saturated chains with since these correspond to permutations of n, we get the unique non-zero value n!. Thus, the invariant delta functions are the divided powers and we may write any invariant function as where =. This gives a natural isomorphism between the reduced incidence algebra and the ring of exponential generating functions. The zeta function is with Möbius function:
Indeed, this computation with formal power series proves that Many combinatorial counting sequences involving subsets or labeled objects can be interpreted in terms of the reduced incidence algebra, and computed using exponential generating functions.

Divisor poset and Dirichlet series

Consider the poset D of positive integers ordered by divisibility, denoted The reduced incidence algebra consists of functions invariant under multiplication, for all For an invariant function, f depends only on b/a, so a natural basis consists of invariant delta functions defined by if b/a = n and 0 otherwise: any invariant function can be written
The product of two invariant delta functions is:

since the only non-zero term comes from c = na and b = mc = nma. Thus, we get an isomorphism from the reduced incidence algebra to the ring of formal Dirichlet series by sending to so that f corresponds to
The incidence algebra zeta function ζD = 1 corresponds to the classical Riemann zeta function having reciprocal where is the classical Möbius function of number theory. Many other arithmetic functions arise naturally within the reduced incidence algebra, and equivalently in terms of Dirichlet series. For example, the divisor function is the square of the zeta function, a special case of the above result that counts the number of elements in the interval ; equivalenty,
The product structure of the divisor poset facilitates the computation of its Möbius function. Unique factorization into primes implies D is isomorphic to an infinite Cartesian product, with the order given by coordinatewise comparison: , where is the kth prime, corresponds to its sequence of exponents Now the Möbius function of D is the product of the Möbius functions for the factor posets, computed above, giving the classical formula:


The product structure also explains the classical Euler product for the zeta function. The zeta function of D corresponds to a Cartesian product of zeta functions of the factors, computed above as so that where the right side is a Cartesian product. Applying the isomorphism which sends t in the kth factor to, we obtain the usual Euler product.

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: