Abel-Grassmann's Groupoids of Modulo Matrices

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).