A non-regular language of infinite trees that is recognizable by a sort-wise finite algebra

TL;DR

This paper demonstrates that for infinite trees, there exists a finite, finitely generated $ ext{omega}$-clone that recognizes a non-regular language, challenging the straightforward algebraic characterization of regularity.

Contribution

It introduces a specific $ ext{omega}$-clone that is finite on each sort and finitely generated yet recognizes a non-regular language, revealing limitations in algebraic characterizations.

Findings

Existence of a finite on each sort, finitely generated $ ext{omega}$-clone recognizing a non-regular language.

Challenges the algebraic characterization of regular languages for infinite trees.

Highlights differences between infinite words and infinite trees in algebraic recognition.

Abstract

$\omega$-clones are multi-sorted structures that naturally emerge as algebras for infinite trees, just as $\omega$-semigroups are convenient algebras for infinite words. In the algebraic theory of languages, one hopes that a language is regular if and only if it is recognized by an algebra that is finite in some simple sense. We show that, for infinite trees, the situation is not so simple: there exists an $\omega$-clone that is finite on every sort and finitely generated, but recognizes a non-regular language.