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1 results

  • Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond

  • Silvio Ghilardi, Luigi Santocanale
  • Journal:
    Mathematical Structures in Computer Science / Volume 30 / Issue 6 / June 2020
    Published online by Cambridge University Press:
    21 January 2020, pp. 572-596

  • 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 fN+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.