Left Transitive AG-groupoids

TL;DR

This paper explores the properties of left transitive AG-groupoids, showing their equivalence with T2-AG-groupoids, and provides a method to identify them, revealing their structural similarities with other algebraic systems.

Contribution

It establishes the equivalence between left transitive AG-groupoids and T2-AG-groupoids and introduces a simple test for identifying such groupoids.

Findings

Left transitive AG-groupoids are equivalent to T2-AG-groupoids.

A simple procedure to test if a groupoid is left transitive AG-groupoid.

Various properties like flexibility and commutativity are equivalent in these structures.

Abstract

An AG-groupoid is an algebraic structure that satisfies the left invertive law: (ab)c =(cb)a. We prove that the class of left transitive AG-groupoids (AG-groupoids satisfying the identity, ab.ac = bc) coincides with the class of T2-AG-groupoids. We also develop a simple procedure to test whether an arbitrary groupoid is left transitive AG-groupoid or not. Further we prove that, (i). Every left transitive AG-groupoid is transitively commutative AG-groupoid (ii) For left transitive AG-groupoid the properties of flexibility, right alternativity, AG*, right nuclear square, middle nuclear square and commutative semigroup are equivalent.