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September 2020 Classes of non-commutative semigroups with finite Fibonacci invariants
Marjan Monsef, Hossein Doostie
Tbilisi Math. J. 13(3): 153-159 (September 2020). DOI: 10.32513/tbilisi/1601344905

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.

Citation

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Marjan Monsef. Hossein Doostie. "Classes of non-commutative semigroups with finite Fibonacci invariants." Tbilisi Math. J. 13 (3) 153 - 159, September 2020. https://doi.org/10.32513/tbilisi/1601344905

Information

Received: 2 February 2020; Accepted: 10 August 2020; Published: September 2020
First available in Project Euclid: 29 September 2020

MathSciNet: MR4154841
Digital Object Identifier: 10.32513/tbilisi/1601344905

Subjects:
Primary: 20M05
Secondary: 20M17

Rights: Copyright © 2020 Tbilisi Centre for Mathematical Sciences

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Vol.13 • No. 3 • September 2020
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