Probabilistic equational spectrum, primality and approximation in finite algebras

TL;DR

This paper introduces a probabilistic framework for analyzing equations in finite algebras, exploring the spectrum of equation probabilities and their relation to algebraic primality and approximation capacity.

Contribution

It defines the probabilistic spectrum of an algebra, studies its properties, and introduces a quantitative primality measure linked to the spectrum's structure.

Findings

The probabilistic spectrum relates to algebraic primality notions.

A primality measure $ ext{Prim}( extbf{A})$ characterizes approximation capacity.

Non-primal two-element algebras have primality at most 1/2.

Abstract

We define the probability of an equation in a finite algebra as the proportion of tuples in its domain that satisfy it. We call the probabilistic spectrum of an algebra the set of probability values obtained when the equation varies. We study fundamental properties of this spectrum, such as density and limit points, and show that its structure is related to several notions of primality of an algebra. We introduce a quantitative measure of primality $\Prim(\A)\in[0,1]$ that characterizes the functional approximation capacity. We show that the degree of primality is related to the size of the spectrum. We also prove that all non-primal two-element algebras satisfy the universal bound $\Prim(\A)\le 1/2$.