Lattice isomorphisms of bisimple monogenic orthodox semigroups

TL;DR

This paper proves that bisimple orthodox semigroups generated by inverse elements of infinite order are uniquely determined by their subsemigroup lattice, extending known results about the bicyclic semigroup.

Contribution

It establishes that such semigroups are strongly lattice determined, generalizing previous findings about the bicyclic semigroup.

Findings

Bisimple orthodox semigroups generated by inverse elements of infinite order are strongly lattice determined.

The result extends the known lattice determination of the bicyclic semigroup.

Provides a classification-based proof of the lattice isomorphism property.

Abstract

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.