Hopfian and co-hopfian subsemigroups and extensions

TL;DR

This paper explores how properties called hopficity and co-hopficity are preserved or not when passing to certain subsemigroups and extensions, revealing nuanced behaviors in finitely generated and finitely presented cases.

Contribution

It provides new examples and results showing that hopficity and co-hopficity are not always preserved under various semigroup extensions and subsemigroup formations.

Findings

Hopficity is not preserved in finitely presented cases.

Co-hopficity is preserved under finite Rees index extensions for finitely generated semigroups.

Without finite generation, co-hopficity is not preserved in extensions.

Abstract

This paper investigates the preservation of hopficity and co-hopficity on passing to finite-index subsemigroups and extensions. It was already known that hopficity is not preserved on passing to finite Rees index subsemigroups, even in the finitely generated case. We give a stronger example to show that it is not preserved even in the finitely presented case. It was also known that hopficity is not preserved in general on passing to finite Rees index extensions, but that it is preserved in the finitely generated case. We show that, in contrast, hopficity is not preserved on passing to finite Green index extensions, even within the class of finitely presented semigroups. Turning to co-hopficity, we prove that within the class of finitely generated semigroups, co-hopficity is preserved on passing to finite Rees index extensions, but is not preserved on passing to finite Rees index subsemigroups, even in the finitely presented case. Finally, by linking co-hopficity for graphs to co-hopficity for semigroups, we show that without the hypothesis of finite generation, co-hopficity is not preserved on passing to finite Rees index extensions.