The monoid of injective and extensive partial transformations of a chain with three elements is nonfinitely based

TL;DR

This paper proves that the monoid of all injective and extensive partial transformations of a three-element chain cannot be described by any finite set of algebraic identities, solving a longstanding problem in the theory of transformation monoids.

Contribution

It establishes that this specific monoid is nonfinitely based, completing the finite basis problem for a key class of partial transformation monoids.

Findings

The monoid admits no finite basis of identities.

The result completes the finite basis problem for a class of partial transformation monoids.

It confirms the nonfinite basis property for the monoid of injective and extensive partial transformations of a three-element chain.

Abstract

We show that the monoid of all injective and extensive partial transformations of a chain with three elements admits no finite basis of its identities. This completes solving of the finite basis problem for the monoids in the basic frame of partial transformation monoids posed by Volkov.