# On the Wilson Monoid of a Pairwise Balanced Design

## TL;DR

This paper explores the algebraic structure called Wilson monoid associated with pairwise balanced designs, linking matroid theory and simplicial complexes, and provides detailed analysis of specific cases like Steiner triple systems.

## Contribution

It introduces the Wilson monoid concept for pairwise balanced designs and investigates its properties, connecting matroid theory with simplicial complexes.

## Key findings

- Wilson monoid W(X) has specific algebraic properties.
- Detailed analysis of Steiner triple systems up to 19 points.
- Study of designs with a block larger than two elements.

## Abstract

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