On the insertion of n-powers

TL;DR

This paper explores the algebraic structure of languages closed under inserting n-powers, comparing related pseudovarieties and identifying the generated pseudovariety of monoids, with implications for regular language closure properties.

Contribution

It characterizes the pseudovariety generated by the inequality 1 ≤ x^n and relates it to known pseudovarieties, providing a simple upper bound and pseudoidentity conditions.

Findings

Identifies the pseudovariety generated by 1 ≤ x^n.

Provides an upper bound satisfying provable pseudoidentities.

Connects the algebraic structure to closure properties of regular languages.

Abstract

In algebraic terms, the insertion of $n$-powers in words may be modelled at the language level by considering the pseudovariety of ordered monoids defined by the inequality $1\le x^n$. We compare this pseudovariety with several other natural pseudovarieties of ordered monoids and of monoids associated with the Burnside pseudovariety of groups defined by the identity $x^n=1$. In particular, we are interested in determining the pseudovariety of monoids that it generates, which can be viewed as the problem of determining the Boolean closure of the class of regular languages closed under $n$-power insertions. We exhibit a simple upper bound and show that it satisfies all pseudoidentities which are provable from $1\le x^n$ in which both sides are regular elements with respect to the upper bound.