{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T14:27:55Z","timestamp":1762352875168,"version":"3.40.5"},"reference-count":29,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2020,1,21]],"date-time":"2020-01-21T00:00:00Z","timestamp":1579564800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2020,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Ruitenburg\u2019s Theorem says that every endomorphism<jats:italic>f<\/jats:italic>of a finitely generated free Heyting algebra is ultimately periodic if<jats:italic>f<\/jats:italic>fixes all the generators but one. More precisely, there is<jats:italic>N<\/jats:italic>\u2265 0 such that<jats:italic>f<\/jats:italic><jats:sup><jats:italic>N<\/jats:italic>+2<\/jats:sup>=<jats:italic>f<\/jats:italic><jats:sup><jats:italic>N<\/jats:italic><\/jats:sup>, 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.<\/jats:p>","DOI":"10.1017\/s0960129519000203","type":"journal-article","created":{"date-parts":[[2020,1,21]],"date-time":"2020-01-21T09:05:56Z","timestamp":1579597556000},"page":"572-596","source":"Crossref","is-referenced-by-count":3,"title":["Free Heyting algebra endomorphisms: Ruitenburg\u2019s Theorem and beyond"],"prefix":"10.1017","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6449-6883","authenticated-orcid":false,"given":"Silvio","family":"Ghilardi","sequence":"first","affiliation":[]},{"given":"Luigi","family":"Santocanale","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2020,1,21]]},"reference":[{"key":"S0960129519000203_ref25","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-59533-6_30"},{"key":"S0960129519000203_ref21","first-page":"217","volume-title":"Resolution of Equations in Algebraic Structures","volume":"1","author":"Goguen","year":"1989"},{"key":"S0960129519000203_ref23","doi-asserted-by":"publisher","DOI":"10.3166\/jancl.17.317-346"},{"key":"S0960129519000203_ref19","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-9936-8"},{"key":"S0960129519000203_ref15","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2003.11.010"},{"key":"S0960129519000203_ref11","first-page":"446","volume-title":"RAMICS","volume":"8428","author":"Frittella","year":"2014"},{"key":"S0960129519000203_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-49630-5_8"},{"key":"S0960129519000203_ref26","doi-asserted-by":"publisher","DOI":"10.2307\/2274142"},{"key":"S0960129519000203_ref9","doi-asserted-by":"publisher","DOI":"10.4169\/000298909X474909"},{"key":"S0960129519000203_ref6","unstructured":"Czarnecki, M. (2010). How fast can the fixpoints in modal \u03bc-calculus be reached? In: Santocanale, L. (ed.) 7th Workshop on Fixed Points in Computer Science, FICS 2010, Brno, Czech Republic, 89. 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(2018). Ruitenburg\u2019s theorem via duality and bounded bisimulations. 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