Sets of universal sequences for the symmetric group and analogous semigroups

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 Sym$(X)$ when $X$ is infinite, and Sierpi\'nski 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 $|Y| \geq |X|$, then every universal sequence for Sym$(X)$ is also universal for Sym$(Y)$. If $|X| > 2 ^ {\aleph_0}$, 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 satisfy this condition, and hence are universal for $X ^ X$ for every infinite set $X$.