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Semi-Heyting Algebras and Identities of Associative Type

dc.contributor.authorCornejo, Juan M.
dc.contributor.authorSankappanavar, Hanamantagouda P.
dc.date.accessioned2019-10-13T10:26:04Z
dc.date.available2019-10-13T10:26:04Z
dc.date.issued2019
dc.identifier.issn0138-0680
dc.identifier.urihttp://hdl.handle.net/11089/30601
dc.description.abstractAn algebra A = ⟨A, ∨, ∧, →, 0, 1⟩ is a semi-Heyting algebra if ⟨A, ∨, ∧, 0, 1⟩ is a bounded lattice, and it satisfies the identities: x ∧ (x → y) ≈ x ∧ y, x ∧ (y → z) ≈ x ∧ [(x ∧ y) → (x ∧ z)], and x → x ≈ 1. SH denotes the variety of semi-Heyting algebras. Semi-Heyting algebras were introduced by the second author as an abstraction from Heyting algebras. They share several important properties with Heyting algebras. An identity of associative type of length 3 is a groupoid identity, both sides of which contain the same three (distinct) variables that occur in any order and that are grouped in one of the two (obvious) ways. A subvariety of SH is of associative type of length 3 if it is defined by a single identity of associative type of length 3. In this paper we describe all the distinct subvarieties of the variety SH of asociative type of length 3. Our main result shows that there are 3 such subvarities of SH.en_GB
dc.description.sponsorshipConsejo Nacional de Investigaciones Cientificas y Tecnicasen_GB
dc.description.sponsorshipUniversidad Nacional del Suren_GB
dc.language.isoenen_GB
dc.publisherWydawnictwo Uniwersytetu Łódzkiegoen_GB
dc.relation.ispartofseriesBulletin of the Section of Logic; 2
dc.rightsThis work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.en_GB
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/4.0en_GB
dc.subjectsemi-Heyting algebraen_GB
dc.subjectHeyting algebraen_GB
dc.subjectidentity of associative typeen_GB
dc.subjectsubvariety of associative typeen_GB
dc.titleSemi-Heyting Algebras and Identities of Associative Typeen_GB
dc.typeArticleen_GB
dc.page.number117–135
dc.contributor.authorAffiliationDepartamento de Matemática, Universidad Nacional del Sur, Bahía Blanca, Argentina
dc.contributor.authorAffiliationDepartment of Mathematics, State University of New York, New Paltz, U.S.A.
dc.identifier.eissn2449-836X
dc.referencesJ. C. Abbott, Semi-Boolean algebras, Matematicki Vesnik 19 (1967), pp. 177–198.en_GB
dc.referencesM. Abad, J. M. Cornejo and J. P. Díaz Varela, Free-decomposability in varieties of semi-Heyting algebras, Mathematical Logic Quarterly 58(3) (2012), pp. 168–176. https://doi.org/10.1002/malq.201020092en_GB
dc.referencesM. Abad, J. M. Cornejo and J. P. Díaz Varela, Semi-Heyting algebras term-equivalent to Gödel algebras, Order 30 (2013), pp. 625–642. http://dx.doi.org/10.1007/s11083-012-9266-0en_GB
dc.referencesM. Abad, J. M. Cornejo and J. P. Díaz Varela, The variety generated by semi-Heyting chains, Soft Computing 15(4) (2010), pp. 721–728. https://doi.org/10.1007/s00500-010-0604-0en_GB
dc.referencesM. Abad, J. M. Cornejo and J. P. Díaz Varela, The variety of semi-Heyting algebras satisfying the equation (0 → 1)*∨(0→ 1)**≈ 1, Reports on Mathematical Logic 46 (2011), pp. 75–90. http://dx.doi.org/10.4467/20842589RM.11.005.0283en_GB
dc.referencesR. Balbes and P.H. Dwinger, Distributive lattices, University of Missouri Press, Columbia, 1974.en_GB
dc.referencesG. Birkhoff, Lattice Theory, First Edition, Colloq. Publ., vol. 25, Providence, 1948.en_GB
dc.referencesG. Birkhoff, Lattice Theory, Third Edition, Colloq. Publ., vol. 25, Providence, 1967.en_GB
dc.referencesS. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Graduate Texts in Mathematics 78, Springer-Verlag, New York, 1981.en_GB
dc.referencesD. Castaño and J. M. Cornejo, Gentzen-style sequent calculus for semi-intuitionistic logic, Studia Logica 104(6) (2016), pp. 1245–1265. https://doi.org/10.1007/s11225-016-9675-yen_GB
dc.referencesJ. M. Cornejo, Semi-intuitionistic logic, Studia Logica 98(1-2) (2011), pp. 9–25. https://doi.org/10.1007/s11225-011-9321-7en_GB
dc.referencesJ. M. Cornejo, The semi-Heyting Brouwer logic, Studia Logica 103(4) (2015), pp. 853–875. https://doi.org/10.1007/s11225-014-9596-6en_GB
dc.referencesJ. M. Cornejo and I. D. Viglizzo, On some semi-intuitionistic logics, Studia Logica 103(2) (2015), pp. 303–344. https://doi.org/10.1007/s11225-014-9568-xen_GB
dc.referencesJ. M. Cornejo and H. P. Sankappanavar, Implication zroupoids and identities of associative type, Quasigroups and Associated Systems 26 (2018), pp. 13–34.en_GB
dc.referencesJ. M. Cornejo and H. J. San Martín, A categorical equivalence between semi-Heyting algebras and centered semi-Nelson algebras, Logic Journal of the IGPL 26(4) (2018), pp. 408–428. https://doi.org/10.1093/jigpal/jzy006en_GB
dc.referencesJ. M. Cornejo and I. Viglizzo, Semi-intuitionistic logic with strong negation, Studia Logica 106(2) (2018), pp. 281–293. https://doi.org/10.1007/s11225-017-9737-9en_GB
dc.referencesJ. M. Cornejo and I. Viglizzo, Semi-Nelson algebras, Order 35(1) (2018), pp. 23–45. https://doi.org/10.1007/s11083-016-9416-xen_GB
dc.referencesA. Day, Varieties of Heyting algebras, I. and II. Preprints.en_GB
dc.referencesV. Glivenko, Sur quelques points de la logique de Brouwer, Académie Royale de Belgique, Bulletins de la Classe des Sciences 15 (1929), pp. 183–188.en_GB
dc.referencesG. Grätzer, Lattice Theory. First Concepts and Distributive Lattices, Freeman, San Francisco, 1971.en_GB
dc.referencesT. Hect and T. Katrinák, Equational classes of relative Stone algebras, Notre Dame Journal of Formal Logic 13 (1972), pp. 248–254. https://doi.org/10.1305/ndjfl/1093894723en_GB
dc.referencesA. Horn, Logic with truth values in a linearly ordered Heyting algebras, Journal of Symbolic Logic 34 (1969), pp. 395–408. https://doi.org/10.2307/2270905en_GB
dc.referencesM. Hosszú, Some functional equations related with the associative law, Publicationes Mathematicae Debrecen 3 (1954), pp. 205–214.en_GB
dc.referencesT. Katriňák, Remarks on the W. C. Nemitz's paper ``Semi-Boolean lattices'', Notre Dame Journal Formal Logic 11 (1970), pp. 425–430. https://doi.org/10.1305/ndjfl/1093894072en_GB
dc.referencesP. Köhler, Brouwerian semilattices, Transactions of the American Mathematical Society 268 (1981), pp. 103–126. https://doi.org/10.1090/S0002-9947-1981-0628448-3en_GB
dc.referencesM. A. Kazim and M. Naseeruddin, On almost-semigroups, The Aligarh Bulletin of Mathematics 2 (1972), pp. 1–7.en_GB
dc.referencesA. A. Monteiro, Axiomes independents pour les algebres de Brouwer, Revista de la Unión Matemática Argentina y de la Asociación Física Argentina} 17 (1955), pp. 148–160.en_GB
dc.referencesW. Nemitz, Implicative semilattices, Transactions of the American Mathematical Society 117 (1965), pp. 128–142. https://doi.org/10.1090/S0002-9947-1965-0176944-9en_GB
dc.referencesW. Nemitz, Semi-Boolean lattices, Notre Dame Journal Formal Logic 10 (1969), pp. 235–238. https://doi.org/10.1305/ndjfl/1093893707en_GB
dc.referencesD. I. Pushkashu, Para-associative groupoids, Quasigroups and Related Systems 18(2) (2010), pp. 187–194.en_GB
dc.referencesH. Rasiowa and R. Sikorski, The Mathematics of metamathematics, PWN, Warsaw, 1969.en_GB
dc.referencesH. Rasiowa, An algebraic approach to non-classical logics, North-Holland, Amsterdam, 1974.en_GB
dc.referencesH. P. Sankappanavar, Heyting algebras with dual pseudocomplementation, Pacific Journal of Mathematics 117 (1985), pp. 405–415. https://doi.org/10.2140/pjm.1985.117.405en_GB
dc.referencesH. P. Sankappanavar, Pseudocomplemented Okham and De Morgan algebras, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 32 (1986), pp. 385–396. https://doi.org/10.1002/malq.19860322502en_GB
dc.referencesH. P. Sankappanavar, Semi-Heyting algebras, Abstracts of Papers Presented to the American Mathematical Society, January 1987, p.13.en_GB
dc.referencesH. P. Sankappanavar, Heyting algebras with a dual lattice endomorphism, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 33 (1987), pp. 712–724. https://doi.org/10.1002/malq.19870330610en_GB
dc.referencesH. P. Sankappanavar, Semi-De Morgan algebras, Journal Symbolic Logic 52 (1987), pp. 712–724. https://doi.org/10.2307/2274359en_GB
dc.referencesH. P. Sankappanavar, Semi-Heyting Algebras: An abstraction from Heyting algebras, Actas del IX Congreso Dr. Antonio A.R. Monteiro (2007), pp. 33–66.en_GB
dc.referencesH. P. Sankappanavar, Expansions of semi-Heyting algebras: Discriminator varieties, Studia Logica 98(1-2) (2011), pp. 27–81. https://doi.org/10.1007/s11225-011-9322-6en_GB
dc.referencesS. K. Stein, On the foundations of quasigroups, Transactions of the American Mathematical Society 85 (1957), pp. 228–256. https://doi.org/10.1090/S0002-9947-1957-0094404-6en_GB
dc.referencesM. H. Stone, Topological representations of distributive lattices and Brouwerian logics, Časopis pro Pěstování Matematiky a Fysiky 67 (1937), pp. 1–25.en_GB
dc.referencesA. Suschkewitsch, On a generalization of the associative law, Transactions of the American Mathematical Society 31(1) (1929), pp. 204–214. https://doi.org/10.1090/S0002-9947-1929-1501476-0en_GB
dc.contributor.authorEmailjmcornejo@uns.edu.ar
dc.contributor.authorEmailsankapph@newpaltz.edu
dc.identifier.doi10.18778/0138-0680.48.2.03
dc.relation.volume48en_GB
dc.subject.msc03G25
dc.subject.msc06D20
dc.subject.msc06D15
dc.subject.msc08B26
dc.subject.msc08B15


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