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A universal sequence for a group or semigroup $S$ is a sequence of words$w_1, w_2, \ldots$ such that for any sequence $s_1, s_2, \ldots\in S$, theequations $w_n = s_n$, $n\in \mathbb{N}$, 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 universalfor the symmetric group Sym$(X)$ when $X$ is infinite, and Sierpi\'nski showedthat $(a ^ 2 b ^ 3 (abab ^ 3) ^ {n + 1} ab ^ 2 ab ^ 3)_{n\in \mathbb{N}}$ isuniversal for the monoid $X ^ X$ of functions from the infinite set $X$ toitself.  In this paper, we show that under some conditions, the set of universalsequences for the symmetric group on an infinite set $X$ is independent of thecardinality 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 universalfor Sym$(Y)$. If $|X| > 2 ^ {\aleph_0}$, then the converse also holds. It isshown that an analogue of this theorem holds in the context of inversesemigroups, where the role of the symmetric group is played by the symmetricinverse monoid. In the general context of semigroups, the full transformationmonoid $X ^ X$ is the natural analogue of the symmetric group and the symmetricinverse monoid. If $X$ and $Y$ are arbitrary infinite sets, then it is an openquestion as to whether or not every sequence that is universal for $X ^ X$ isalso universal for $Y ^ Y$. However, we obtain a sufficient condition for asequence to be universal for $X ^ X$ which does not depend on the cardinalityof $X$. A large class of sequences satisfy this condition, and hence areuniversal for $X ^ X$ for every infinite set $X$.

Sets of universal sequences for the symmetric group and analogous semigroups