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Minimal probabilistic spectrum grupoids

Carles Cardó

Abstract

The equational probabilistic spectrum of a Finite algebra is defined as the set of probabilities with which equations are satisfied in the algebra. We investigate algebras A with minimal spectrum, that is, spectra consisting only of the values 1 and 1/|A|. We focus in particular on groupoids and semigroups. We prove that, apart from trivial cases, groupoids with minimal spectrum are quasigroups. Moreover, we show that weak associativity conditions force such quasigroups to be groups. Finally, we give a complete structural classification of semigroups with minimal spectrum.

Minimal probabilistic spectrum grupoids

Abstract

The equational probabilistic spectrum of a Finite algebra is defined as the set of probabilities with which equations are satisfied in the algebra. We investigate algebras A with minimal spectrum, that is, spectra consisting only of the values 1 and 1/|A|. We focus in particular on groupoids and semigroups. We prove that, apart from trivial cases, groupoids with minimal spectrum are quasigroups. Moreover, we show that weak associativity conditions force such quasigroups to be groups. Finally, we give a complete structural classification of semigroups with minimal spectrum.
Paper Structure (6 sections, 13 theorems, 56 equations)

This paper contains 6 sections, 13 theorems, 56 equations.

Table of Contents

  1. 1. Introduction
  2. 2. The equational probabilistic spectrum of an algebra
  3. 3. Minimality of the spectrum and quasigroups
  4. 4. Minimal spectrum semigroups
  5. 5. Weak associativity
  6. 6. Small groupoids with minimal spectrum

Key Result

Proposition 2.4

Let $\mathcal{A}$ and $\mathcal{B}$ be algebras.

Theorems & Definitions (28)

  • Definition 2.1
  • proof
  • Example 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Lemma 3.6
  • proof
  • Theorem 3.7
  • proof
  • ...and 18 more