The Size of Generating Sets of Powers

TL;DR

This paper establishes a dichotomy for finite algebras, showing they either have polynomially generated powers or exponentially generated powers, with a simple criterion for the latter in idempotent cases.

Contribution

It proves a dichotomy for finite algebras regarding their generating powers and provides a simple criterion for EGP in idempotent algebras.

Findings

Finite algebras have either PGP or EGP property.

A simple criterion for EGP in idempotent algebras is provided.

The dichotomy is proven for all finite algebras.

Abstract

In the paper we prove for every finite algebra A that either it has the polynomially generated powers (PGP) property, or it has the exponentially generated powers (EGP) property. For idempotent algebras we give a simple criteria for the algebra to satisfy EGP property.