Obtaining a group from distributive lattices
Let $P$ be a finite lattice with incidence algebra $I(P,K)$ over a field $K$. A function $f$ in $I(P,K)$ is called strongly multiplicative if $f(x \land y , x \lor y)=f(x \land y, x) f(x \land y ,y )$ for all $x,y \in P$. $P$ is distributive if and only if the strongly multiplicative invertible functions form a group.
My question is whether there is an easy way to obtain this group using Sage for a given $P$.
One can of course implement this group. It has a lot of elements, even if $K$ is finite, so the implementation depends on the needs. What should be done with this sage implementation? It is a good piece of work...