,O)$, where we identify multiplication by 0 as giving the empty function. It is straightforward then to see that the ideal $I(X)$ of partial constant maps of $W(X)$ is isomorphic to the Rees matrix semigroup~\cite[A4]{qtheory} $\mathcal{M}^{0}(\{1\}, S,\mathcal{O},<,>)$.
We now note that the natural left action of $W(X)$ on $S$, that is, $fp =f(p), f \in W(X), p \in S$ by partial functions has an ``adjoint'' right action on $\mathcal{O}$. Namely, let $f \in W(X)$ and define the partial function $\bar{f}:\mathcal{O} \rightarrow \mathcal{O}$ acting on the right of $\mathcal{O}$ by $O\bar{f}=f^{-1}(O)$ if this set is non-empty and undefined otherwise. Adjointness means that for the pairing $<,>$ defined above, we have ${ \; \neq\; \; \neq\; $. If we think of $<,>$ as a $|S| \times |\mathcal{O}|$ matrix over $\{0,1\}$, then reduced means that distinct rows (columns) are not equal to one another. 0-simple semigroups over the trivial group with reduced structure matrices are precisely the congruence-free 0-simple semigroups and along with the semigroups of order 2 and finite simple groups, form the class of all finite congruence-free semigroups~\cite[Theorem~4.7.17]{qtheory}. A semigroup $S$ is called Generalized Group Mapping (GGM) if it has a unique 0-minimal ideal $I(S)$ which is a 0-simple semigroup and such that $S$ acts faithfully on both the left and right of $I(S)$ by left and right multiplication~\cite[Chapter~4]{qtheory}. More precisely, $S$ acts faithfully by partial functions on any $\mathcal{L}$ and $\mathcal{R}$ class in $I(S) \setminus \{0\}$, which means both the left and right Sch{\"u}tzenberger representations on the $\mathcal{J}$-class $I(S) \setminus \{0\}$ are faithful. An important theorem says that if the maximal subgroup of $I(S) \setminus \{0\}$ is trivial, then $S$ is GGM if and only if $I(S)$ is a congruence-free 0-simple semigroup and $S$ is a subsemigroup of the translational hull of $I(S)$~\cite[Sections~4.6, 5.5]{qtheory}. We summarize all of this discussion in the following Theorem. By ``non-trivial $PBD$'' we mean one that contains at least two blocks, that is, it is not the $PBD$ $(S,\{S\})$.
\begin{theo}
Let $X=(S,\mathcal{L})$ be a non-trivial $PBD$ and $W(X)$ its Wilson monoid of continuous functions. Let $\mathcal{O}$ be the collection of non-empty open subsets of $X$. Then $W(X)$ is a GGM semigroup with unique 0-minimal ideal $I(X)$ isomorphic to the congruence-free Rees matrix semigroup $\mathcal{M}^{0}(\{1\}, S,\mathcal{O},<,>)$, where $$ (since any two points are in exactly one block and we are assuming that there are at least two blocks) and for each $O \neq O' \in \mathcal{O}$, there is a $p \in S$ such that $