The Catalan monoid $C_5$ is inherently nonfinitely based relative to finite Jtrivial semigroups
05/12/2017 05.12.2017 room 6.2.33, FCUL  Dept. Matemática
Mikhail V. Volkov (Ural Federal University, Russia)
Faculdade de Ciências da Universidade de Lisboa
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 6element 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 Jtrivial 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 orderpreserving and decreasing transformations of the 5element chain is INFB relative to varieties generated by finite Jtrivial 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 $B_2^1$).
