A nonfinitely based semigroup of triangular matrices

TL;DR

This paper demonstrates that the semigroup of all upper triangular real 3x3 matrices with 0s and 1s on the diagonal does not have a finite identity basis, extending previous conditions to this specific case.

Contribution

It applies a new sufficient condition for nonfinitely based semigroups to the semigroup of upper triangular matrices, showing it lacks a finite identity basis.

Findings

The semigroup $ ext{UT}_3( ext{R})$ has no finite identity basis.

The result also holds as an involution semigroup under reflection.

Extends the nonfinitely based property to a specific matrix semigroup.

Abstract

A new sufficient condition under which a semigroup admits no finite identity basis has been recently suggested in a joint paper by Karl Auinger, Yuzhu Chen, Xun Hu, Yanfeng Luo, and the author (see http://arxiv.org/abs/1405.0783). Here we apply this condition to show the absence of a finite identity basis for the semigroup $\mathrm{UT}_3(\mathbb{R})$ of all upper triangular real $3\times 3$-matrices with 0s and/or 1s on the main diagonal. The result holds also for the case when $\mathrm{UT}_3(\mathbb{R})$ is considered as an involution semigroup under the reflection with respect to the secondary diagonal.