A Factorization Theorem for Forest Algebras

2026-05-11

AI Summary

This work addresses the challenge of tracking subforest positions in recursive decompositions of forest structures by proposing a bounded-depth decomposition method inspired by Simon’s factorization theorem. Within the framework of forest algebras, the approach extends free forest algebras and introduces a novel semantic constraint—$\mathcal{R}$-alignment—that ensures semantic compatibility across different decomposition strategies at the semigroup level. The study establishes that $\mathcal{R}$-alignment is both necessary and sufficient for the existence of bounded-depth decompositions: under this condition, every homomorphism admits such a decomposition, whereas counterexamples exist in its absence. This contribution provides a systematic algebraic decomposition theory for forest structures, significantly extending the applicability of algebraic automata and semigroup techniques to tree-like data.

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).

Problem

• forest algebras
• Simon's factorization theorem
• bounded-depth decomposition
• morphisms
• R-alignment

Innovation

• forest algebras
• Simon's factorization theorem
• R-alignment
• bounded-depth decomposition
• algebraic automata theory