# Epimorphisms, dominions and H-commutative semigroups

## TL;DR

This paper investigates the structural properties of H-commutative semigroups, generalizes key results from commutative semigroup theory, and extends the concept of saturation to this broader class of semigroups.

## Contribution

It generalizes Isbell's result on dominions and extends the saturation property to H-commutative semigroups satisfying the minimum condition.

## Key findings

- The dominion of an H-commutative semigroup is H-commutative.
- H-commutative semigroups satisfying the minimum condition are saturated.
- Provides new examples and structural insights into H-commutative semigroups.

## Abstract

In the present paper, a series of results and examples that explore the structural features of H-commutative semigroups are provided. We also generalise a result of Isbell from commutative semigroups to H-commutative semigroups by showing that the dominion of an H-commutative semigroup is H-commutative. We then use this to generalise Howie and Isbell's result that any H-commutative semigroup satisfying the minimum condition on principal ideals is saturated.