Generators and presentations of inverse subsemigroups of the monogenic free inverse semigroup

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 $E(S)$.