Ideals in intra-regular left almost semigroups

TL;DR

This paper introduces and characterizes (1,2)-ideals in intra-regular LA-semigroups, showing their equivalence with two-sided ideals and exploring various ideal properties and conditions for intra-regularity.

Contribution

It defines (1,2)-ideals in LA-semigroups and establishes their equivalence with two-sided ideals in intra-regular cases, providing comprehensive ideal characterizations.

Findings

(1,2)-ideals coincide with two-sided ideals in intra-regular LA-semigroups

Characterization of intra-regular LA-semigroups via ideal properties

Equivalent conditions for two-sided ideals in intra-regular LA-semigroups

Abstract

In this paper, we have introduced the notion of (1,2)-ideal in an LA-semigroup and shown that (1,2)-ideal and two-sided ideal coincide in an intra-regular LA-semigroup. We have characterized an intra-regular LA-semigroup by using the properties of left and right ideals. Some natural examples of LA-semigroups have been given. Further we have investigated some useful conditions for an LA-semigroup to become an intra-regular LA-semigroup and given the counter examples to illustrate the converse inclusions. All the ideals (left, right, two-sided, interior, quasi, bi- generalized bi- and (1,2)) of an intra-regular LA-semigroup have been characterized. Finally we have given an equivalent statement for a two-sided ideal of an intra-regular LA-semigroup in terms of the intersection of two minimal two-sided ideals of an intra-regular LA-semigroup.