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Semiring and involution identities of powers of inverse semigroups

Igor Dolinka, Sergey V. Gusev, Mikhail V. Volkov

Abstract

The set of all subsets of any inverse semigroup forms an involution semiring under set-theoretical union and element-wise multiplication and inversion. We find structural conditions on a finite inverse semigroup guaranteeing that neither semiring nor involution identities of the involution semiring of its subsets admit a finite identity basis.

Semiring and involution identities of powers of inverse semigroups

Abstract

The set of all subsets of any inverse semigroup forms an involution semiring under set-theoretical union and element-wise multiplication and inversion. We find structural conditions on a finite inverse semigroup guaranteeing that neither semiring nor involution identities of the involution semiring of its subsets admit a finite identity basis.
Paper Structure (4 sections, 12 theorems, 7 equations)

This paper contains 4 sections, 12 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Powers of finite Clifford semigroups
  3. Inherently nonfinitely based ai-semi-rings and involution semigroups
  4. Proofs of Theorems \ref{['thm:powerinvsgp']} and \ref{['thm:invpowerinvsgp']}

Key Result

Theorem 1.1

Let $\mathcal{S}=(S,\cdot)$ be a finite inverse semigroup. Suppose that either $\mathcal{S}$ is not Clifford or all subgroups of $\mathcal{S}$ are solvable and at least one of them is nonabelian. Then the identities of the power semiring $(\mathcal{P}(S);\,\cup,\cdot)$ admit no finite basis.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.2: Putcha:79
  • Proposition 2.3
  • proof
  • Remark 1
  • Proposition 3.1
  • Remark 2
  • ...and 8 more