Strongly nonfinitely based monoids

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. This gives a first example of a strongly nonfinitely based monoid that is not inherently nonfinitely based monoid, thus solving a 20+ years old problem. The result has numerous implications for the finite basis problem for finite semigroups.

This a joint work with Sergey Gusev and Olga Sapir.