On semigroups and groupoids with minimal probabilistic spectrum

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.