Ideals in Left Almost Semigroups

TL;DR

This paper explores the properties of ideals in left almost semigroups, establishing conditions under which ideals are prime, semiprime, and forming ordered structures, thereby advancing the algebraic understanding of LA-semigroups.

Contribution

It characterizes prime ideals in LA-semigroups with various conditions, linking ideal properties to order structures and irreducibility, which is a novel algebraic insight.

Findings

Prime ideals are characterized by idempotency and total order of ideals.

In regular LA-semigroups, prime ideals correspond to total order of ideals.

Ideals are prime iff they are strongly irreducible and ideals form a semilattice.

Abstract

A left almost semigroup (LA-semigroup) or an Abel-Grassmann's groupoid (AG-groupoid) is investigated in several papers. In this paper we have discussed ideals in LA-semigroups. Specifically, we have shown that every ideal in an LA-semigroup S with left identity e is prime if and only if it is idempotent and the set of ideals of S is totally ordered under inclusion. We have shown that an ideal of S is prime if and only if it is semiprime and strongly irreducible. We have proved also that every ideal in a regular LA-semigroup S is prime if and only if the set of ideals of S is totally ordered under inclusion. We have proved in the end that every ideal in S is prime if and only if it is strongly irreducible and the set of ideals of S form a semilattice.