Inverse semigroups with certain types of partial automorphism monoids

Abstract

AbstractFor an inverse semigroupS, the set of all isomorphisms betweeninverse subsemigroups ofSis an inverse monoid under composition which is denoted by(S) and called the partial automorphism monoid ofS. Kirkwood [7] and Libih [8] determined which groups have Clifford partial automorphism monoids. Here we investigate the structure of inverse semigroups whose partial automorphism monoids belong to certain other important classes of inverse semigroups. First of all, we describe (modulo so called “exceptional” groups) all inverse semigroupsSsuch that(S) is completely semisimple. Secondly, for an inverse semigroupS, we find a convenient description of the greatest idempotent-separating congruence on(S), using a well-known general expression for this congruence due to Howie, and describe all those inverse semigroups whose partial automorphism monoids are fundamental.