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.

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, 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

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

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.

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(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.

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