A partial semigroup (or PSG) consists of a set $X$ and of a partial composition law $*$ defined over $X$, that is to say:(1) $x*y$ is not always defined,(2) if $(x*y)*z$ is defined, so is $x*(y*z)$, and the two values are equal,(3) if $x*(y*z)$ is defined, so is $(x*y)*z$, and the two values are equal.Let $\mathbb{R}$ be a ring. We may then define a convolution product over $\mathbb{R}^X$ : given two functions $f,g$, we define a function $f*g$ such thatfor every $w \in X$, $f*g(w) = \sum_{x,y \in X : x*y = w} f(x) g(y)$.By a graded PSG, we mean a pair $(S,r)$ consisting of a PSG $S = (X,*)$ and of a rank function $r : X \rightarrow \mathbb{N}$ such that whenever $x*y$ is defined, $r(x*y) = r(x)+r(y)$. This holds for instance for the interval algebra of an Eulerian poset (see On the Foundations of Combinatorial Theory I. Theory of Moebius Functions, by G.C. Rota).By analogy with the situation in posets, we may define a 'triangular matrix' $\lambda \in \mathbb{R}^X$ by $\lambda(x) = 1$ for every $x \in X$, and a mapping $\zeta : \mathbb{R}^X \rightarrow \mathbb{R}^X$ by $\zeta(x) = \lambda * x$. It is known that for a partial semigroup obtained from the interval algebra of an Eulerian poset, the function $\lambda$ has an inverse $\mu \in \mathbb{R}^X$ given by $\mu([x,y]) = (-1)^{r(y)-r(x)}$. This suggests a possible extension to certain graded PSGs, which leads me to ask the following.Question: for a graded PSG, under what conditions does $\lambda$ have an inverse $\mu$? When this holds, is it always true that $\mu(x) = (-1)^{r(x)}$?(NOTE: I renamed the rank function $r$ instead of $l$ for greater consistency.)
Convolution products in partial semigroups
order theory