Ruitenburg’s Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ulti- mately 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 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.
Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond / S. Ghilardi, L. Santocanale. - In: MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE. - ISSN 0960-1295. - (2020). [Epub ahead of print] [10.1017/S0960129519000203]
Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond
S. Ghilardi;
2020
Abstract
Ruitenburg’s Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ulti- mately 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 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.File | Dimensione | Formato | |
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