Certain numerical results in non-associative structures

Abstract

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 nth-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 = yx 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 nth-commutativity degree of two infinite classes of finite loops, which are non-commutative and non-associative. Also by deriving explicit expressions for nth-commutativity degree of these loops, we will obtain best upper bounds for this probability.

On Generalized Relative Commutativity Degree of Finite Moufang Loop

“…A quasi-group is a nonempty set with a binary operation such that for every three elements x, y and z of that, the equation xy = z has a unique solution in this set, whenever two of the three element are specified. A quasi-group with a neutral element is called a loop and following [2,6,7,8] one may see the definition of Moufang loop satisfying four tantamount relators. These loops are of interest because of their appearance in the projective geometry as planes and even they are non-associative, they retain many properties of the groups.…”

“…These loops are studied for their finiteness property in [1,2]. It is obvious that M (G, 2) is non-associative if and only if the group G is non-abelian.…”