Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond

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 ≥ 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.

An Algebraic Approach to Inquisitive and -Logics

“…This property however is shared in a more general form by every polynomial . Ruitenberg’s Theorem [23, 43] states that for any polynomial we can find a number such that . This allows to introduce the -variant of an intermediate logic L as and to generalize our study of -logics to arbitrary -variants.…”