Nilpotent inverse semigroups with central idempotents

Abstract:

An inverse semigroup $S$ with central idempotents, i.e. a strong semilattice of groups, will be called nilpotent, if it is finite and if for each prime divisor ${p_i}$ of the orders of the structure groups of $S$ the sets ${P_i} = \{ s \in S|o(s) = p_i^{{k_s}}, {k_s} \geqslant 0\}$ are subsemigroups of $S$. If $S$ is a group, then ${P_i}$ are exactly the Sylow ${p_i}$-subgroups of the group. A theory similar to that given by W. Burnside for finite nilpotent groups is developed introducing the concepts of ascending resp. descending central series in an inverse semigroup, and it is shown that almost all of the well-known properties of finite nilpotent groups do hold also for the class of finite inverse semigroups with central idempotents.