Numerical invariants of identities of unital algebras

TL;DR

This paper investigates how adjoining an external unit to unital algebras affects their polynomial identity exponents, revealing that for many algebras it increases by exactly 1 and that any real number in [2,3] can be realized as such an exponent.

Contribution

It demonstrates that adjoining an external unit can increase the PI-exponent by 1 for many algebras and constructs unital algebras with any PI-exponent in [2,3], expanding understanding of polynomial identities.

Findings

PI-exponent increases by 1 when external unit is adjoined for many algebras

Any real number in [2,3] can be realized as a PI-exponent of some unital algebra

Adjoining external units influences the polynomial identity structure of algebras

Abstract

We study polynomial identities of algebras with adjoined external unit. For a wide class of algebras we prove that adjoining external unit element leads to increasing of PI-exponent precisely to 1. We also show that any real number from the interval [2,3] can be realized as PI-exponent of some unital algebra.