Ideals in Left Almost Semigroups
Qiaser Mushtaq, Madad Khan

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsFuzzy and Soft Set Theory
