Enumeration of finite inverse semigroups

We give an efficient algorithm for the enumeration up to isomorphism of the inverse semigroups of order n, and we count the number S(n) of inverse semigroups of order $$n\le 15$$. This improves considerably on the previous highest-known value S(9). We also give a related algorithm for the enumeration up to isomorphism of the finite inverse semigroups S with a given underlying semilattice of idempotents E, a given restriction of Green’s $$\mathrel {\mathcal {D}}$$-relation on S to E, and a given list of maximal subgroups of S associated to the elements of E.