On Free $\omega$-Continuous and Regular Ordered Algebras

TL;DR

This paper characterizes free ordered $ ext{Sigma}$-algebras with restricted completeness and continuity, introducing quasi-regular varieties and showing how free regular algebras relate to free $ ext{omega}$-continuous algebras through subalgebras of coterms.

Contribution

It provides a general framework for understanding free algebras in quasi-regular varieties, connecting regular and $ ext{omega}$-continuous algebras via submonads of coterms.

Findings

Free regular algebras are subalgebras of free $ ext{omega}$-continuous algebras.

Characterization of free algebras using submonads of coterms.

Applicability to various algebraic structures like Kleene algebras and semirings.

Abstract

We study varieties of certain ordered $\Sigma$-algebras with restricted completeness and continuity properties. We give a general characterization of their free algebras in terms of submonads of the monad of $\Sigma$-coterms. Varieties of this form are called \emph{quasi-regular}. For example, we show that if $E$ is a set of inequalities between finite $\Sigma$-terms, and if $\mathcal{V}_\omega$ and $\mathcal{V}_\mathrm{reg}$ denote the varieties of all $\omega$-continuous ordered $\Sigma$-algebras and regular ordered $\Sigma$-algebras satisfying $E$, respectively, then the free $\mathcal{V}_\mathrm{reg}$-algebra $F_\mathrm{reg}(X)$ on generators $X$ is the subalgebra of the corresponding free $\mathcal{V}_\omega$-algebra $F_\omega(X)$ determined by those elements of $F_\omega(X)$ denoted by the regular $\Sigma$-coterms. This is a special case of a more general construction that applies to any quasi-regular family. Examples include the *-continuous Kleene algebras, context-free languages, $\omega$-continuous semirings and $\omega$-continuous idempotent semirings, OI-macro languages, and iteration theories.