  In this paper we study the regular semigroups weakly generated by a single
element x, that is, with no proper regular subsemigroup containing x. We show
there exists a regular semigroup $F_1$ weakly generated by x such that all
other regular semigroups weakly generated by x are homomorphic images of $F_1$.
We define $F_1$ using a presentation where both sets of generators and
relations are infinite. Nevertheless, the word problem for this presentation is
decidable. We describe a canonical form for the congruence classes given by
this presentation, and explain how to obtain it. We end the paper studying the
structure of $F_1$. In particular, we show that the `free regular semigroup
$FI_2$ weakly generated by two idempotents "is isomorphic to a regular
subsemigroup of $F_1$ weakly generated by {xx',x'x}.
