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

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.