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Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond

Published online by Cambridge University Press:  21 January 2020

Silvio Ghilardi Open the ORCID record for Silvio Ghilardi [Opens in a new window]  and
Luigi Santocanale
Silvio Ghilardi*
Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano
Luigi Santocanale
Affiliation:
LIS, CNRS UMR 7020, Aix-Marseille Université
*
*Corresponding author. Email: silvio.ghilardi@unimi.it
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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 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.

Keywords

Heyting algebraRuitenburg’s Theoremsheaf dualitybounded bisimulationsfree algebra endomorphisms

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 30 , Special Issue 6: Special Issue: Unification , June 2020 , pp. 572 - 596
Copyright
© Cambridge University Press 2020

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