# The algebra of Boolean matrices, correspondence functors, and simplicity

## TL;DR

This paper calculates the dimensions of all simple modules for the algebra of Boolean matrices, linking it to simple correspondence functors and employing lattice theory and functor methods.

## Contribution

It provides a complete determination of simple module dimensions for Boolean matrix algebras using advanced functor and lattice techniques.

## Key findings

- Dimensions of all simple modules are explicitly determined.
- Establishes a connection between Boolean matrix algebra and correspondence functors.
- Introduces new tools in finite lattice theory for algebraic analysis.

## Abstract

We determine the dimension of every simple module for the algebra of the monoid of all relations on a finite set (i.e. Boolean matrices). This is in fact the same question as the determination of the dimension of every evaluation of a simple correspondence functor. The method uses the theory of such functors developed in [BT2, BT3], as well as some new ingredients in the theory of finite lattices.