The Finite Basis Problem for Kauffman Monoids

TL;DR

This paper establishes a condition that prevents certain semigroups, including Kauffman monoids for n≥3, from having a finite set of identities, advancing understanding of their algebraic structure.

Contribution

It provides a sufficient condition for semigroups to lack a finite identity basis and applies this to prove Kauffman monoids are nonfinitely based for all n≥3.

Findings

Kauffman monoids for n≥3 are nonfinitely based.

The result applies to involution semigroup structures of Kauffman monoids.

A general sufficient condition for nonfinite basis in semigroups is established.

Abstract

We prove a sufficient condition under which a semigroup admits no finite identity basis. As an application, it is shown that the identities of the Kauffman monoid $\mathcal{K}_n$ are nonfinitely based for each $n\ge 3$. This result holds also for the case when $\mathcal{K}_n$ is considered as an involution semigroup under either of its natural involutions.