[Paper Review] On some classes of Abel-Grassmann's groupoids

Madad Khan, Faisal Faisal+1 more|arXiv (Cornell University)|Oct 28, 2010

semigroups and automata theory4 references4 citations

TL;DR

This paper investigates structural equivalences among various regularity classes—weakly regular, intra-regular, right regular, left regular, left quasi regular, and completely regular—in Abel-Grassmann groupoids (AG-groupoids) with a left identity. It proves that all these regularity notions coincide in such structures, establishing a complete equivalence. The key contribution is the unification of these regularity concepts under a single framework, showing that in AG-groupoids with left identity, all forms of regularity are equivalent and imply the groupoid is equal to its square (S = S²).

ABSTRACT

In this paper, we have investigated different classes of an AG-groupoid by their structural properties. We have prove that weakly regular, intra-regular, right regular, left regular, left quasi regular and completely regular coincide in an AG-groupoid with left identity and in AG^{**}-groupoid. Further we have prove that every regular (weakly regular, intra-regular, right regular, left regular, left quasi regular, completely regular) AG-groupoid with left identity (AG^{**}-groupoid) is regular but the converse is not true in general. Also it has been shown that non-associative regular, weakly regular, intra-regular, right regular, left regular, left quasi regular and completely regular AG^{*} groupoids do not exist.

Motivation & Objective

• To investigate the structural properties of different regularity classes in Abel-Grassmann groupoids (AG-groupoids).
• To determine the conditions under which various regularity notions—weakly regular, intra-regular, right regular, left regular, left quasi regular, and completely regular—coincide.
• To clarify the relationship between regularity and the algebraic structure S = S² in AG-groupoids.
• To examine whether regularity concepts can be defined in AG∗-groupoids, which satisfy the identity (ab)c = b(ac).

Proposed method

• Utilizing the left invertive law (ab)c = (cb)a and medial law (ab)(cd) = (ac)(bd) as foundational identities in AG-groupoids.
• Applying the paramedial law (ab)(cd) = (dc)(ba) and the left identity property a(bc) = b(ac) in AG-groupoids with left identity.
• Proving equivalences between regularity classes through algebraic manipulation using the identities (1)–(4) and Theorem 1.
• Constructing counterexamples to show that S = S² does not imply regularity, demonstrating the converse is not true.
• Using permutation-based identities in AG∗-groupoids to show that regularity implies commutative semigroup structure.
• Establishing equivalence chains via lemmas and theorems, such as showing weak regularity implies right and left regularity through substitution and identity application.

Experimental results

Research questions

• RQ1 Do weakly regular, intra-regular, right regular, left regular, left quasi regular, and completely regular AG-groupoids coincide in the presence of a left identity?
• RQ2 Is the condition S = S² sufficient for regularity in AG-groupoids?
• RQ3 Can regularity concepts be consistently defined in AG∗-groupoids?
• RQ4 What is the relationship between weak regularity and other regularity classes in AG-groupoids with left identity?
• RQ5 Under what conditions does S = S² imply that an AG-groupoid is regular?

Key findings

• In an AG-groupoid with left identity, all regularity classes—weakly regular, intra-regular, right regular, left regular, left quasi regular, and completely regular—are equivalent.
• The paper proves that S = S² holds for all regularity classes in AG-groupoids with left identity, but the converse is not true in general.
• A counterexample is provided showing that S = S² does not imply regularity, as demonstrated by an AG-groupoid on six elements where S = S² but elements like d are not regular.
• In AG-groupoids with left identity, weak regularity implies right regularity and left regularity via algebraic identities and substitutions.
• All regularity classes in AG-groupoids with left identity imply regularity, and the converse is not true, as shown by the failure of (ax)a = (ax)(ay) unless additional conditions like idempotency or commutativity hold.
• In AG∗-groupoids, any form of regularity (weak, intra, right, left, left quasi, completely) implies the structure becomes a commutative semigroup, as S = S² and the permutation identity (6) enforce commutativity.