the nth commutativity degree of semigroups

For a given positive integer n, the nth commutativity degree of a finite noncommutative semigroup s is defined to be the probability of choosing a pair (x, y) for x, y ∈ s such that xn and y commute in s. if for every elements x and y of an associative algebraic structure (s, .) there exists a positive integer r such that xy = yrx, then s is called quasi-commutative. evidently, every abelian group or commutative semigroup is quasi-commutative. in this paper, we study the nth commutativity degree of certain classes of quasi-commutative semigroups. we show that the nth commutativity degree of such structures is greater than 1/ 2 . finally, we compute the nth commutativity degree of a finite class of non-quasi-commutative semigroups and we conclude that it is less than 1/ 2 .