# On inverse ordered semigroups

## TL;DR

This paper explores the structure of inverse ordered semigroups, generalizing inverse semigroups to include order, and provides characterizations based on their idempotents and regularity.

## Contribution

It introduces the concept of inverse ordered semigroups and characterizes them through their regularity, inverse properties, and idempotent elements.

## Key findings

- An ordered semigroup is a complete semilattice of t-simple ordered semigroups if and only if it is completely regular and inverse.
- Inverse ordered semigroups are characterized by their ordered idempotents.
- The paper establishes equivalences and structural properties of inverse ordered semigroups.

## Abstract

The purpose of this paper is to study the generalization of inverse semigroups (without order). An ordered semigroup S is called an inverse ordered semigroup if for every a 2 S, any two inverses of a are H-related. We prove that an ordered semigroup is complete semilattice of t-simple ordered semigroups if and only if it is completely regular and inverse. Furthermore characterizations of inverse ordered semigroups have been characterized by their ordered idempotents.