Idempotent ordered semigroup

TL;DR

This paper explores the structure of idempotent ordered semigroups, defining ordered idempotents and examining their properties, including their relation to semilattices and rectangular idempotent semigroups.

Contribution

It introduces the concept of ordered idempotents and characterizes idempotent ordered semigroups as semilattices of rectangular idempotent semigroups.

Findings

Every idempotent semigroup forms a complete semilattice of rectangular idempotent semigroups.

The paper establishes fundamental properties of ordered idempotents within semigroups.

Connections between idempotent ordered semigroups and other algebraic structures are discussed.

Abstract

An element e of an ordered semigroup $(S,\cdot,\leq)$ is called an ordered idempotent if $e\leq e^2$. We call an ordered semigroup $S$ idempotent ordered semigroup if every element of $S$ is an ordered idempotent. Every idempotent semigroup is a complete semilattice of rectangular idempotent semigroups and in this way we arrive to many other important classes of idempotent ordered semigroups.