The incidence functions $I(P,R)$ of a locally finite poset and their convolution

Let $(P,\le )$ be a locally finite poset (Intervals in a poset; locally finite, lower-finite and upper-finite posets) and let $R$ be a commutative ring (Commutative ring, Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides). Put

$\mathrm{Int}(P):=\{(x,y)\in P\times P:x\le y\}\mathrm{.}$

An incidence function with coefficients in $R$ is a function $f:\mathrm{Int}(P)\to R$. The set of all incidence functions is denoted

$I(P,R):=R^{\mathrm{Int}(P)}\mathrm{.}$

Addition, zero and additive inverses are pointwise, as in the function ring of The ring $R^X$ of all functions from a set $X$ into a ring, with pointwise operations. For $f,g\in I(P,R)$ their convolution is the incidence function

$(f\ast g)(x,y):={\sum }_{z\in [x,y]}f(x,z)g(z,y)(x\le y),$

where the sum is the finite commutative-monoid sum of A finite sum in a commutative monoid indexed by an arbitrary finite set in the additive monoid of $R$.

This operation is well defined precisely at the stated level of generality: local finiteness makes $[x,y]$ finite for each comparable pair, so the displayed ring-valued sum has finitely many terms. The definition makes no claim about sums over an entire principal ideal or principal filter.