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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.
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Contributor : Luigi Santocanale <>
Submitted on : Thursday, January 3, 2019 - 7:33:27 PM
Last modification on : Friday, November 29, 2019 - 8:15:43 AM
Document(s) archivé(s) le : Thursday, April 4, 2019 - 4:07:15 PM


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  • HAL Id : hal-01969235, version 1
  • ARXIV : 1901.01252



Silvio Ghilardi, Luigi Santocanale. Free Heyting Algebra Endomorphisms: Ruitenburg's Theorem and Beyond. Mathematical Structures in Computer Science, Cambridge University Press (CUP), In press. ⟨hal-01969235⟩



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