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Semiring identities in the semigroup $B_2$

Vyacheslav Yu. Shaprynskiǐ

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.

Semiring identities in the semigroup $B_2$

Abstract

The 5-element Brandt semigroup 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.

Paper Structure

This paper contains 2 sections, 9 theorems, 46 equations.

Table of Contents

  1. Introduction and summary
  2. Proofs of Theorems \ref{['main1']} and \ref{['main2']}

Key Result

Theorem 1

The following identities constitute an identity basis for the semiring $B_2$ within the variety of all ai-semirings: where $x_1,x_2,z_1,z_2$ can be empty. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 6 more