Semiring identities of the Brandt monoid

TL;DR

This paper investigates the algebraic structure of the 6-element Brandt monoid, showing its addition operation is a term operation of an inverse semigroup and proving its semiring identities are not finitely based.

Contribution

It demonstrates that the unique addition making the Brandt monoid an idempotent semiring is a term operation of its inverse semigroup structure, and provides a simple proof of the non-finite basis of its semiring identities.

Findings

Addition is a term operation of the inverse semigroup

Semiring identities of the Brandt monoid are not finitely based

Simplified proof of non-finite basis of identities

Abstract

The 6-element Brandt monoid $B_2^1$ admits a unique addition under which it becomes an additively idempotent semiring. We show that this addition is a term operation of $B_2^1$ as an inverse semigroup. As a consequence, we exhibit an easy proof that the semiring identities of $B_2^1$ are not finitely based.