Classes of non-commutative semigroups with finite Fibonacci invariants

Abstract

For a $2$-generated semigroup $S$ we define the invariant $\lambda (S)$ as the minimum of the Fibonacci lengths over all generating pairs of the semigroup $S$, where the Fibonacci length with respect to a generating pair $(x, y)$ is the fundamental period (if exist) of the truncated periodic sequence $z_0=x, z_1=y, z_k=z_{k-2}z_{k-1}, (k\geq 2)$ of the elements of $S$. We name this invariant as the Fibonacci invariant of $S$. Our used notation is the same as of the celebrated work of D.L. Johnson in $2005$ on infinite groups. In this paper we examine two classes of semigroups for existence of this invariant. The considered semigroups are the finite semigroup $S=\langle a, b\mid a^{p^\alpha}=a, b^{q^\beta}=b,ab=a\rangle$ of order $p^{\alpha}q^{\beta}-1$ and the infinite semigroup $T=\langle a, b\mid a^{p^\alpha}=a, ab=a\rangle$, for all integers $\alpha, \beta \geq 2$ and all distinct primes $p$ and $q$. We prove the existence of the Fibonacci invariants of these semigroups, for all parameters. As a numerical result we show that $\lambda (T)\leq \lambda (S)=p^{\alpha +1}$, if $p$ is even.