The Catalan monoid $C_5$ is inherently nonfinitely based relative to finite J-trivial semigroups

Recall that a finite semigroup $S$ is said to be inherently nonfinitely based (INFB) if $S$ does not belong to any finitely based locally finite variety. In 1987, Mark Sapir proved that the 6-element Brandt monoid $B_2^1$ is INFB; later he gave an algorithmically efficient description of INFB semigroups. Sapir's description implies, in particular, that no finite J-trivial semigroup is INFB.

The concept of an INFB semigroup has been generalized by Jackson and the speaker as follows: given a class $\mathcal{C}$ of semigroup varieties, a finite semigroup $S$ is said to be INFB relative to $\mathcal{C}$ if $S$ does not belong to any finitely based variety from $\mathcal{C}$. ("Classic" INFB semigroups are just semigroups which are INFB relative to locally finite varieties.)

In the talk we present a fresh result by Olga Sapir and the speaker that the Catalan monoid $C_5$, that is, the monoid of all order-preserving and decreasing transformations of the 5-element chain is INFB relative to varieties generated by finite J-trivial semigroups. The proof relies on Simon's celebrated theorem on piecewise testable languages.

We also survey some other interesting equational properties of $C_5$ which are surprisingly similar to the properties of $B_2^1$: this part of the talk is based on recent results by Klima, Kunz, and Polak (for $C_5$) and Jackson (for $B_2^1$).