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 $1 + \frac{B(n+2)-B(n+1)+B(n)}2$ for $n\geq2$, 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.

💡 Research Summary

The paper “Minimum transformation representations of diagram monoids” determines the exact minimum transformation degree—i.e., the smallest size of a full transformation monoid into which a given finite monoid embeds—for all the most frequently studied families of finite diagram monoids. These families include the partition monoid (P_n), the partial Brauer monoid (PB_n), the Brauer monoid (B_n), the planar partition monoid (PP_n), the Motzkin monoid (M_n) and the Temperley–Lieb monoid (TL_n).

The authors begin by recalling that any finite monoid (M) embeds in a full transformation monoid (\mathcal{T}_n) (the monoid of all functions on an (n)-element set) and define the minimum transformation degree (\deg(M)) as the smallest such (n). They also introduce the minimum partial‑transformation degree (\deg’(M)=\deg(M)-1), which corresponds to faithful actions on a set that contains a global fixed point. While the existence of an embedding is guaranteed by a Cayley‑type theorem, determining the exact minimal (n) is notoriously difficult, especially for monoids with rich Green’s relations such as diagram monoids.

A central technical contribution of the paper is a unified framework based on regular (*)-semigroups and partial actions on projections. Every diagram monoid under consideration is a regular ()-semigroup; its set of projections (P(S)={p\in S\mid p^2=p=p^}) can be identified with the set of idempotents that are fixed by the involution. The authors show that the natural partial action of a regular (*)-semigroup on its projections is faithful precisely when the underlying right congruence contains no non‑trivial two‑sided congruence. This observation allows them to replace the problem of finding a minimal faithful transformation representation by the problem of finding a small subset (Q\subseteq P(S)) that still yields a faithful action.

For the partition monoid (P_n) they construct a specific subset (Q) consisting of projections of “low rank”. By a careful combinatorial count they prove
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