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. ‎If an integer‎ ‎be $k\geq 2$‎, ‎let $S$ will be a finite $k$-generated as well as non-associative algebraic structure $S= $‎, ‎where‎ ‎$A=\lbrace a_{1}‎, ‎a_{2},\dots‎, ‎a_{k}\rbrace$‎, ‎and the sequence‎ ‎$$x_{i}=\left\{‎ ‎\begin{array}{ll}‎ ‎a_{i}‎, ‎& 1\leq i\leq k‎, ‎\\‎ ‎x_{i-k}(x_{i-k+1}(\ldots(x_{i-3}(x_{i-2}x_{i-1}))\ldots))‎, ‎& i>k‎, ‎\end{array}‎ ‎\right‎. ‎$$‎ ‎is called the $k$-nacci sequence of $S$ with respect to the generating set $A$‎, ‎as denoted in $k_{A}(S)$‎. ‎When $k_{A}(S)$ is periodic‎, ‎we will use the length of the period of the periodicity length of $S$ proportional to $A$ in $LEN_{A}(S)$‎ ‎and the minimum of the positive integers of $LEN_A(S)$ will be mentioned as periodicity invariant of $S$‎, ‎denoted in $\lambda_k(S)$‎. ‎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 ‎$x_i=x_{i-k}x_{i-k+1}\dots x_{i-3}x_{i-2}x_{i-1}$‎, ‎for every $i\geq k+1$‎. ‎Thus‎, ‎we attempt to give explicit upper‎ ‎bounds for the periodicity invariant of two infinite classes of‎ ‎finite non-associative $3$-generated algebraic structures‎. ‎Moreover‎, ‎two classes of non-isomorphic Moufang loops of the same periodicity length were obtained in the study‎.