Periodicity invariant of finitely generated algebraic structures
DOI:
https://doi.org/10.30495/jme.v0i0.1266Keywords:
Non-associativity, Loops, Periodic sequences, Finitely generated structures.Abstract
In this paper, we discuss the periodicity problems in the finitely generated algebraic structures and exhibit their naturalsources 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 $S=<A>$, 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.
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