Strongly nonfinitely based monoids

TL;DR

This paper demonstrates that a specific 42-element monoid cannot be generated by any finitely based finite semigroup, highlighting limitations in the algebraic structure of certain monoids.

Contribution

It proves that the monoid of partial order preserving and extensive injections on a 4-element chain is not contained in any variety generated by a finitely based finite semigroup.

Findings

The 42-element monoid is not contained in any finitely based variety.

The monoid exhibits properties that defy finite basis generation.

This provides a counterexample in the theory of monoid varieties.

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 semigroup.