Structure of a finite commutative inverse semigroup and a finite band for which the inverse monoid of local automorphisms is permutable. (Q1759986)

A semigroup \(S\) is called permutable if any two congruences on \(S\) commute with respect to composition. A local automorphism of a semigroup \(S\) is an isomorphism between two subsemigroups of \(S\). The set \(PA(S)\) of all local automorphisms of \(S\) is an inverse submonoid of the full symmetric inverse monoid on \(S\). The main results of this paper are the following two theorems: Theorem 2. Let \(S\) be a commutative inverse semigroup. Then the semigroup \(PA(S)\) is permutable if and only if \(S\) is either a linearly ordered semilattice or a primitive semilattice or an elementary Abelian \(p\)-group. Theorem 3. Let \(S\) be a finite band. Then the semigroup \(PA(S)\) is permutable if and only if \(S\) is either a linearly ordered semilattice or a primitive semilattice or a semigroup of one-sided zeros.