Certain numerical results in non-associative structures

The finite non-commutative and non-associative algebraic structures are indeed one of the special structures for their probabilistic results in some branches of mathematics. For a given integer n ≥ 2 , the n th-commutativity degree of a finite algebraic structure S , denoted by P n ( S ) , is the probability that for chosen randomly two elements x and y of S , the relator x n y = y x n holds. This degree is specially a recognition tool in identifying such structures and studied for associative algebraic structures during the years. In this paper, we study the n th-commutativity degree of two infinite classes of finite loops, which are non-commutative and non-associative. Also by deriving explicit expressions for n th-commutativity degree of these loops, we will obtain best upper bounds for this probability.