Certain numerical results in non-associative structures

Behnam Azizi,Hossein Doostie

doi:10.1007/s40096-018-0274-0

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 nge 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^ny=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.

Highlights

A quasigroup is a non-empty 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 the set, whenever two of the three elements are specified
A quasi-group with a neutral element is called a loop, and following [5, 1,2,3], and one may see the definition of Moufang loop satisfying four tantamount relators
These loops are of interest because they retain main properties of the groups [4, 5]

Summary

Introduction

Let G be a finite group and n ≥ 2 be a positive integer. Our main results are: Proposition 2.1 Let G be a finite non-abelian group and n ≥ 2 be a positive integer. We prove the following results about these classes of loops: Proposition 2.2 Let M = M(D2m, 2) be a finite Moufang loop, where m ≥ 3, and n be an odd integer and d = g.c.d(m, 2n).

Results

Conclusion