Ideals in Left Almost Semigroups

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.

6 pages, 1table

Cited by in corpus (8)

• On some classes of Abel-Grassmann's groupoids
• Soft Neutrosophic Algebraic Structures and Their Generalization, Vol. 2
• Ideals in intra-regular left almost semigroups
• On Generalised Inflations of Right Modular Groupoids
• Characterizations of intra-regular gamma AG-groupoids by the properties of their gamma ideals
• Characterizations of LA-semigroups by new types of fuzzy ideals
• The spectrum of the variety of anti-rectangular Abel Grassmann bands
• A new generalization of fuzzy ideals in LA-semigroups