  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 $X$.
