Free Heyting Algebra Endomorphisms: Ruitenburg's Theorem and Beyond

TL;DR

This paper provides a semantic proof of Ruitenburg's Theorem on the periodicity of endomorphisms in finitely generated free Heyting algebras and explores the conditions under which arbitrary endomorphisms are ultimately periodic.

Contribution

It introduces a semantic proof using duality and bisimulation ranks and investigates the periodicity of general endomorphisms beyond the classical theorem.

Findings

Endomorphisms fixing all but one generator are ultimately periodic with period 2.

Arbitrary endomorphisms are not always ultimately periodic.

Period bounds can be explicitly determined in locally finite subvarieties.

Abstract

Ruitenburg's Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ultimately periodic if f fixes all the generators but one. More precisely, there is N $\ge$ 0 such that f N +2 = f N , thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms between free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators.