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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Sets of universal sequences for the symmetric group and analogous semigroups

Authors: J. Hyde, J. Jonušas, J. D. Mitchell and Y. H. Péresse
Journal: Proc. Amer. Math. Soc. 148 (2020), 1917-1931
MSC (2010): Primary 20B07, 20B30, 20M20
Published electronically: January 28, 2020
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Abstract: A universal sequence for a group or semigroup $ S$ is a sequence of words $ w_1, w_2, \ldots $ such that for any sequence $ s_1, s_2, \ldots \in S$, the equations $ w_n = s_n$, $ n\in \mathbb{N}$, can be solved simultaneously in $ S$. For example, Galvin showed that the sequence $ (a^{-1}(a^nba^{-n})b^{-1}(a^nb^{-1}a^{-n})ba)_{n\in \mathbb{N}}$ is universal for the symmetric group $ \operatorname {Sym}(X)$ when $ X$ is infinite, and Sierpiński showed that $ (a ^2 b ^3 (abab ^3) ^{n + 1} ab ^2 ab ^3)_{n\in \mathbb{N}}$ is universal for the monoid $ X ^X$ of functions from the infinite set $ X$ to itself.

In this paper, we show that under some conditions, the set of universal sequences for the symmetric group on an infinite set $ X$ is independent of the cardinality of $ X$. More precisely, we show that if $ Y$ is any set such that $ \vert Y\vert \geq \vert X\vert$, then every universal sequence for $ \operatorname {Sym}(X)$ is also universal for $ \operatorname {Sym}(Y)$. If $ \vert X\vert > 2 ^{\aleph _0}$, then the converse also holds. It is shown that an analogue of this theorem holds in the context of inverse semigroups, where the role of the symmetric group is played by the symmetric inverse monoid. In the general context of semigroups, the full transformation monoid $ X ^X$ is the natural analogue of the symmetric group and the symmetric inverse monoid. If $ X$ and $ Y$ are arbitrary infinite sets, then it is an open question as to whether or not every sequence that is universal for $ X ^X$ is also universal for $ Y ^Y$. However, we obtain a sufficient condition for a sequence to be universal for $ X ^X$ which does not depend on the cardinality of $ X$. A large class of sequences satisfies this condition, and hence is universal for $ X ^X$ for every infinite set $ X$.

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Additional Information

J. Hyde
Affiliation: Cornell University, Ithaca, New York 14850

J. Jonušas
Affiliation: TU Wien, Karlsplatz 13, 1040 Wien, Austria

J. D. Mitchell
Affiliation: University of St Andrews, St Andrews KY16 9AJ, Scotland

Y. H. Péresse
Affiliation: University of Hertfordshire, Hatfield AL10 9EU, United Kingdom

Received by editor(s): January 17, 2019
Received by editor(s) in revised form: August 9, 2019, September 12, 2019, and September 13, 2019
Published electronically: January 28, 2020
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2020 American Mathematical Society