International Journal of Algebra, Vol. 6, 2012, no. 15, 705 - 712 On Varieties of (n, m)-Semigroups Irena Stojmenovska Orce Nikolov 96 1000 Skopje Republic of Macedonia irena.stojmenovska@gmail.com Donˇco Dimovski Faculty of Natural Sciences & Mathematics Ss. Cyril and Methodius University Skopje P.O. Box 162, 1000 Skopje, Republic of Macedonia donco@pmf.ukim.mk Abstract We investigate varieties of (n, m)-semigroups. A direct descrip- tion of the complete system of (n, m)-identities for a variety of (n, m)- semigroups is obtained. Mathematics Subject Classification: 20M07, 20M10, 20N99 Keywords: (n, m)-semigroup, (n, m)-identity, variety of (n, m)-semigroups 1 Introduction and preliminaries Vector valued, i.e. (m + k, m)-groupoids, semigroups and groups were intro- duced in [9] and [1]. Since then, the theory of (n, m)-structures was developed and interesting results were obtained (see for example [2], [5], [6], [10]). Vector valued algebraic structures are a generalization of the usual binary, i.e. (2, 1) algebraic structures. On the one hand they are similar to the binary structures, but on the other hand they incorporate new ideas and specific properties. Throughout the sequel we assume that Q = ∅, n,m ∈ N and n−m = k ≥ 1. The set of all positive integers will be denoted by N and N 0 = N ∪{0}. Also, N t and N t,0 will denote the sets {1, 2,...,t} and {0, 1, 2,...,t}, t ∈ N. If x =(a 1 ,a 2 ,...,a t ) ∈ Q t (where Q t is the cartesian product of t copies of Q), then we write x = a t 1 , and we identify x with the word a 1 a 2 ...a t . For such an x we say that its length |x| is t. The notation a t r where r>t will be identified