The #CSP problem consists of counting the number of satisfying of assignements for a given CSP. It is conjectured that for each constraint language B, the #CSP(B) problem is either solvable in P-time or #P-complete.
We study the particular case of the problem which requires counting the number of solutions to a system over a fixed finite semigroup. We show that for each semigroup this problem is indeed in FP or #P-complete and provide the precise classification. We also discuss the light shed by these results on the current status of the #CSP-dichotomy conjecture.