\cPA-isomorphisms of inverse semigroups

A partial automorphism of a semigroup $S$ is any isomorphism between its subsemigroups, and the set all partial automorphisms of $S$ with respect to composition is the inverse monoid called the partial automorphism monoid of $S$. Two semigroups are said to be \cPA-isomorphic if their partial automorphism monoids are isomorphic. A class \K of semigroups is called \cPA-closed if it contains every semigroup \cPA-isomorphic to some semigroup from \K. Although the class of all inverse semigroups is not \cPA-closed, we prove that the class of inverse semigroups, in which no maximal isolated subgroup is a direct product of an involution-free periodic group and the two-element cyclic group, is \cPA-closed. It follows that the class of all combinatorial inverse semigroups (those with no nontrivial subgroups) is \cPA-closed. A semigroup is called \cPA-determined if it is isomorphic or anti-isomorphic to any semigroup that is \cPA-isomorphic to it. We show that combinatorial inverse semigroups which are either shortly connected [5] or quasi-archimedean [10] are \cPA-determined

Lattice isomorphisms of bisimple monogenic orthodox semigroups

Using the classification and description of the structure of bisimple monogenic orthodox semigroups obtained in \cite{key10}, we prove that every bisimple orthodox semigroup generated by a pair of mutually inverse elements of infinite order is strongly determined by the lattice of its subsemigroups in the class of all semigroups. This theorem substantially extends an earlier result of \cite{key25} stating that the bicyclic semigroup is strongly lattice determined.Comment: Semigroup Forum (published online: 15 April 2011

Bisimple monogenic orthodox semigroups

We give a complete description of the structure of all bisimple orthodox semigroups generated by two mutually inverse elements

On lattice isomorphisms of orthodox semigroups

Two semigroups are lattice isomorphic if the lattices of their subsemigroups are isomorphic, and a class of semigroups is lattice closed if it contains every semigroup which is lattice isomorphic to some semigroup from that class. An orthodox semigroup is a regular semigroup whose idempotents form a subsemigroup. We prove that the class of all orthodox semigroups in which every nonidempotent element has infinite order is lattice closed