Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond
Silvio Ghilardi, Luigi Santocanale · Mathematical Structures in Computer Science · 2020
Abstract Ruitenburg’s Theorem says that every endomorphismfof a finitely generated free Heyting algebra is ultimately periodic ifffixes all the generators but one. More precisely, there isN≥ 0 such thatfN+2=fN, 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 of 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.