  It was proved by Oliveira and Silva (2005) that every finitely generated
inverse subsemigroup of the monogenic free inverse semigroup $FI_1$ is finitely
presented. The present paper continues this development, and gives generating
sets and presentations for general (i.e. not necessarily finitely generated)
inverse subsemigroups of $FI_1$. For an inverse semigroup $S$ and an inverse
subsemigroup $T$ of $S$, we say $S$ is finitely generated modulo $T$ if there
is a finite set $A$ such that $S = \langle T, A \rangle$. Likewise, we say that
$S$ is finitely presented modulo $T $ if $S$ can be defined by a presentation
of the form $\text{Inv}\langle X, Y \mid R, Q\rangle$, where $\text{Inv}\langle
X\mid R\rangle$ is a presentation for $T$ and $Y$ and $Q$ are finite. We show
that every inverse subsemigroup $S$ of $FI_1$ is finitely generated modulo its
semilattice of idempotents $E(S)$. By way of contrast, we show that when $S\neq
E(S)$, it can never be finitely presented modulo $E(S)$. However, in the
process we establish some nice (albeit infinite) presentations for $S$ modulo
$E(S)$.
