Minimum Transformation Representations of Diagram Monoids

Abstract We obtain formulae for the minimum transformation degrees of the most well-studied families of finite diagram monoids, including the partition, Brauer, Temperley–Lieb, and Motzkin monoids. For example, the partition monoid $\mathcal{P}_{n}$ has degree $1 + \frac{B(n+2)-B(n+1)+B(n)}2$ for $n\geq 2$, where these are Bell numbers. The proofs involve constructing explicit faithful representations of the minimum degree, many of which can be realised as (partial) actions on projections.