On hopfian cofinite subsemigroups

TL;DR

This paper investigates the conditions under which the hopfian property of semigroups is inherited by cofinite subsemigroups, showing that finite generation is crucial for this inheritance.

Contribution

It establishes that finitely generated semigroups with hopfian cofinite subsemigroups are hopfian, and provides counterexamples when the semigroup is not finitely generated.

Findings

Finitely generated semigroups with hopfian cofinite subsemigroups are hopfian.

The property does not hold for non-finitely generated semigroups.

Existence of finitely generated hopfian semigroups with non-hopfian subsemigroups.

Abstract

If a finitely generated semigroup S has a hopfian (meaning: every surjective endomorphism is an automorphism) cofinite subsemigroup T then S is hopfian too. This no longer holds if S is not finitely generated. There exists a finitely generated hopfian semigroup S with a non-hopfian subsemigroup T such that S\T has size 1.