Sets of universal sequences for the symmetric group and analogous semigroups

TL;DR

This paper investigates universal sequences in symmetric groups and semigroups, showing their independence from the set size under certain conditions and providing criteria for universality across different infinite sets.

Contribution

It establishes conditions under which universal sequences are independent of the cardinality of the underlying set and extends results to inverse semigroups and transformation monoids.

Findings

Universal sequences for symmetric groups are independent of the set size when the cardinality is large.

An analogue of the independence result holds for inverse semigroups.

A sufficient condition for universality of sequences in transformation monoids is provided, independent of set size.

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 $X$.