# On inverse and right inverse ordered semigroups

## TL;DR

This paper develops the theory of right inverse ordered semigroups, characterizing their structure and relationships with other types of ordered semigroups, and establishing foundational results in this area.

## Contribution

It introduces the concept of right inverse ordered semigroups and provides characterizations and foundational results for this class of semigroups.

## Key findings

- A regular ordered semigroup is right inverse iff any two right inverses are $
$-related.
- Characterizations of right Clifford, right group-like, and group-like ordered semigroups are provided.
- A foundational framework for right inverse semigroups is established.

## Abstract

A regular ordered semigroup $S$ is called right inverse if every principal left ideal of $S$ is generated by an $\mathcal{R}$-unique ordered idempotent. Here we explore the theory of right inverse ordered semigroups. We show that a regular ordered semigroup is right inverse if and only if any two right inverses of an element $a\in S$ are $\mathcal{R}$-related. Furthermore, different characterizations of right Clifford, right group-like, group like ordered semigroups are done by right inverse ordered semigroups. Thus a foundation of right inverse semigroups has been developed.