periodicity invariant of finitely generated algebraic structures

. in this paper, we discuss the periodicity problems in the finitely generated algebraic structures and exhibit their natural sources in the theory of invariants of finite groups and it forms an interesting and relatively self-contained nook in the imposing edifice of group theory. one of the deepest and important results of the related theory of finite groups is a complete classification of all periodic groups, that is, the finite groups with periodic properties. for every integer k ≥ 2 and a k−generated non-associative algebraic structure s =< a >, where a = {a1, a2, . . . , ak}, the sequence xi =  ai, 1 ≤ i ≤ k, xi−k(xi−k+1(. . .(xi−3(xi−2xi−1)). . .)), i > k, is called the k-nacci sequence of s with respect to the generating set a, denoted ka(s). if ka(s) is periodic, we call the length of the period of the sequence the periodicity length of s with respect to the generating set a, written lena(s) and the minimum of the positive integers of lena(s) will be mentioned as periodicity invariant of s, denoted by λk(s). however, this invariant has been studied for groups and semigroups during the years as well as the associative property of s where above sequence was reduced to xi = xi−kxi−k+1 . . . xi−3xi−2xi−1, forevery i ≥ k + 1. thus, we attempt to give explicit upper bounds for theperiodicity invariant of two infinite classes of finite non-associative 3-generated algebraic structures. moreover, two classes of non-isomorphicmoufang loops of the same periodicity length were obtained in the study.