On intra-regular and left regular and left duo ordered $\Gamma$-semigroups

TL;DR

This paper characterizes the structure of intra-regular and left regular left duo ordered $ ext{Γ}$-semigroups by describing principal filters and relating regularity to the semiprimeness of ideals.

Contribution

It provides a structural description of principal filters and establishes equivalences between regularity conditions and semiprimeness of ideals in ordered $ ext{Γ}$-semigroups.

Findings

Principal filters are key to the structure of intra-regular and left regular left duo ordered $ ext{Γ}$-semigroups.

Intra-regularity is equivalent to all ideals being semiprime.

Left (right) regular and left (right) duo properties correspond to semiprime left (right) ideals.

Abstract

For an intra-regular or a left regular and left duo ordered $\Gamma$-semigroup $M$, we describe the principal filter of $M$ which plays an essential role in the structure of this type of $po$-$\Gamma$-semigroups. We also prove that an ordered $\Gamma$-semigroup $M$ is intra-regular if and only if the ideals of $M$ are semiprime and it is left (right) regular and left (right) duo if and only if the left (right) ideals of $M$ are semiprime.