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.