Abel-Grassmann Groupoids of Modulo Matrices

TL;DR

This paper introduces Matrix AG-groupoids and Matrix AG-groups over modulo integers, exploring their properties such as associativity deviations, transitive commutativity, and conditions for being AG-bands, expanding algebraic structures over modular matrices.

Contribution

It defines new nonassociative algebraic structures called Matrix AG-groupoids and Matrix AG-groups over Zn and investigates their fundamental properties.

Findings

Existence of Matrix AG-groupoids for all n ≥ 3

Matrix AG-groupoids are transitively commutative and cancellative if n is prime

Conditions under which these structures are AG-bands

Abstract

The binary operation of usual addition is associative in all common matrices over R. However, here we define a binary operation of addition in matrices over Zn which present the concept of nonassociativity. These structures form Matrix AG-groupoids and Matrix AG-groups over modulo integers Zn. We show that both these structures exist for every integer n geq 3, and explore some of their properties like: (i). Every matrix AG-groupoid G_n AG(t, u), is transitively commutative AG-groupoid and is a cancellative AG-groupoid if n is prime. (ii). Every matrix AG-groupoid of Type G_AG-II(n) is T3-AG-groupoid. (iii). A matrix AG-groupoid G_nAG(t, u) is an AG-band, if t + u = 1(mod n).