Catalan monoids inherently nonfinitely based relative to finite $\mathcal{R}$-trivial semigroups

Abstract

We show that the 42-element monoid of all partial order preserving and extensive injections on the 4-element chain is not contained in any variety generated by a finitely based finite $\mathcal{R}$-trivial semigroup. This provides unified proofs for several known facts and leads to a bunch of new results on the Finite Basis Problem for finite $\mathcal{R}$- and $\mathcal{J}$-trivial semigroups.

37 pages, 5 figures. In version 3, the reference list is updated and a few typos are fixed