Structure of a finite commutative inverse semigroup and a finite bundle for which the inverse monoid of local automorphisms is permutable - Derech, V. D. (Ukrains’kyi Matematychnyi Zhurnal, 2011)
For a semigroup $S$, the set of all isomorphisms between subsemigroups of $S$ is an inverse monoid with respect to composition, which is denoted by $P A(S)$ and is called the monoid of local automorphisms of $S$. A semigroup $S$ is called permutable if, for any pair of congruences $p, σ$ on $S$, one has $p ° σ = σ ° p$. We describe the structure of a finite commutative inverse semigroup and a finite band whose monoids of local automorphisms are permutable.