Minimum transformation representations of diagram monoids

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 P_n has degree (Formula presented) for n ≥ 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.