The nth commutativity degree of semigroups

For a given positive integer n, the n th 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 x n 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 = y rx, then S is called quasi-commutative. Evidently, every abelian group or commutative semigroup is quasi-commutative. In this paper, we study the n th commutativity degree of certain classes of quasi-commutative semigroups. We show that the n th commutativity degree of such structures is greater than 1 2 . Finally, we compute the n th commutativity degree of a finite class of non-quasi-commutative semigroups and we conclude that it is less than 1 2