On semigroups and groupoids with minimal probabilistic spectrum

TL;DR

This paper investigates finite algebras with minimal probabilistic spectrum, revealing that non-trivial cases are quasigroups, and classifies semigroups with this property.

Contribution

It characterizes algebras with minimal spectrum, showing that certain weak conditions imply full associativity and providing a complete classification of such semigroups.

Findings

Groupoids with minimal spectrum are quasigroups (except trivial cases).

Weak associativity conditions collapse into full associativity.

Complete classification of semigroups with minimal spectrum.

Abstract

The equational probabilistic spectrum of a finite algebra is the set of probabilities with which equations are satisfied in the algebra. We study algebras with minimal spectrum, that is, spectra consisting only of the values $1$ and $1/|\A|$. We show that, apart from trivial cases, groupoids with minimal spectrum are quasigroups. We further prove that several weak associativity conditions collapse into full associativity, and hence into group structure. Finally, we obtain a complete classification of semigroups with minimal spectrum.