Classification of finite nilsemigroups for which the inverse monoid of local automorphisms is permutable semigroup

Abstract

A semigroup $S$ is called permutable if $\rho \circ \sigma = \sigma \circ \rho$ for any pair of congruences $\rho$, $\sigma$ on $S$. A local automorphism of the semigroup $S$ is defined as an isomorphism between two subsemigroups of this semigroup. The set of all local automorphisms of a semigroup $S$ with respect to an ordinary operation of composition of binary relations forms an inverse monoid of local automorphisms. In the proposed paper, we present a classification of all finite nilsemigroups for which the inverse monoid of local automorphisms is permutable. Полугруппа $S$ называется перестановочной, если для любой пары конгруэнций $\rho$, $\sigma$ на $S$ имеет место равенство $\rho \circ \sigma = \sigma \circ \rho$.