Semiring identities in the semigroup $B_2$

TL;DR

This paper establishes a finite basis of identities for the semiring structure of the 5-element Brandt semigroup $B_2$, solving the Finite Basis Problem for a class of semilattice-ordered inverse semigroups.

Contribution

It provides the first finite identity basis for the semiring structure of $B_2$, advancing understanding of algebraic identities in semilattice-ordered inverse semigroups.

Findings

Finite basis of identities for $B_2$ semiring established

Complete solution of the Finite Basis Problem for this class

Semiring structure derived from semilattice-ordered inverse semigroup

Abstract

The 5-element Brandt semigroup $B_2$ admits the structure of a naturally semilattice-ordered inverse semigroup, thus becoming an additively idempotent semiring with the operation of taking greatest lower bounds as the semiring addition. For this semiring we present a finite basis of identities and thus, by the previous results, complete the solution of Finite Basis Problem for combinatorial naturally semilattice-ordered inverse semigroups.