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

TL;DR

This paper proves that certain Catalan monoids are inherently nonfinitely based relative to finite R-trivial semigroups, advancing understanding of the finite basis problem in semigroup theory.

Contribution

It demonstrates that a specific 42-element monoid cannot be generated by any finitely based finite R-trivial semigroup, providing new insights into the finite basis problem.

Findings

The 42-element monoid is not contained in any variety generated by a finitely based finite R-trivial semigroup.

Provides unified proofs for known facts about finite R-trivial semigroups.

Introduces new results on the finite basis problem for R- and J-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.