The finite basis problem for the monoid of 2 by 2 upper triangular tropical matrices

TL;DR

This paper proves that the monoid of 2 by 2 upper triangular tropical matrices cannot be described by a finite set of identities, revealing deep algebraic complexity in tropical matrix monoids.

Contribution

It establishes the nonfinite basis property for the monoid of 2x2 upper triangular tropical matrices, extending results from identity-based monoids.

Findings

Monoid satisfying all identities u_n = v_n for all n is nonfinitely based.

The monoid of 2x2 upper triangular tropical matrices is nonfinitely based.

Results connect tropical algebra with algebraic identity theory.

Abstract

For each positive $n$, let $u_n = v_n$ denote the identity obtained from the Adjan identity $(xy) (yx) (xy) (xy) (yx) = (xy) (yx) (yx) (xy) (yx)$ by substituting $(xy) \rightarrow (x_1 x_2 \dots x_n)$ and $(yx) \rightarrow (x_n \dots x_2 x_1)$. We show that every monoid which satisfies $u_n = v_n$ for each positive $n$ and generates the variety containing the bicyclic monoid is nonfinitely based. This implies that the monoid of 2 by 2 upper triangular tropical matrices over the tropical semiring is nonfinitely based.