A Factorization Theorem for Forest Algebras

TL;DR

This paper extends Simon's factorization theorem from words to forests within forest algebras, introducing a new semantic restriction called -alignment that guarantees bounded-depth decompositions.

Contribution

It develops a recursive factorization framework for forests and identifies -alignment as a key condition for bounded-depth decompositions in forest algebras.

Findings

Introduces -alignment as a semantic restriction ensuring bounded decompositions.

Provides a recursive factorization framework for forests.

Shows that without -alignment, bounded decompositions may not exist.

Abstract

Simon's factorization theorem is a celebrated tool in algebraic automata theory, providing bounded-depth decompositions of words with respect to morphisms into finite semigroups. We develop an analogue of Simon's theorem for \emph{forests} in the setting of forest algebras. In contrast with words, this presents a basic difficulty: recursively factoring a forest requires keeping track of where each subforest ``fits''. This difficulty ripples throughout the proof, and we overcome it by augmenting the free forest algebra and by developing a framework that supports recursive factorization of forests, along with its semantic implications. Our main result identifies a new semantic restriction on morphisms (called $\mathcal{R}$-alignment) which intuitively ensures that different ways of cutting a forest remain compatible (in a certain sense) at the semigroup level. Under this condition, we prove that every morphism admits decompositions of bounded depth. We also prove that without this restriction, there are morphisms for which no bounded-depth decomposition exists (under our notion of decomposition).