  We give a new perspective of the relationship between simple matroids of rank
3 and pairwise balanced designs, connecting Wilson's theorems and tools with
the theory of truncated boolean representable simplicial complexes. We also
introduce the concept of Wilson monoid W(X) of a pairwise balanced design X. We
present some general algebraic properties and study in detail the cases of
Steiner triple systems up to 19 points, as well as the case where a single
block has more than 2 elements
