Mehran University Research Journal Of Engineering &
Technology (HEC Recognized In Category "X")
Publishing Since 1982.



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Abel-Grassmann's Groupoids of Modulo Matrices

Keywords: AG-groupoid and AG-group of Matrices overZ n , T3-AG-groupoid, Transitively Commutative AG-groupoid, Cancellative AG-groupoid.

Mehran University Research Journal of Engineering & Technology

Volume 35 ,  Issue 1

MUHAMMAD  RASHAD,AMANULLAH  ,IMTIAZ  AHMED

Abstract

The binary operation of usual addition is associative in all matrices over R. However, a binary operation of addition in matrices over Z n of a nonassociative structures of AG-groupoids and AG-groups are defined and investigated here. It is shown that both these structures exist for every integer n > 3. Various properties of these structures are explored like: (i) Every AG-groupoid of matrices over Z n is transitively commutative AG-groupoid and is a cancellative AG-groupoid ifn is prime. (ii) Every AG-groupoid of matrices over Z n of Type-II is a T3-AG-groupoid. (iii) An AG-groupoid of matrices over Z n ; G nAG(t,u), is an AG-band, ift+ u=1(mod n).