Representation of Lattices with Modal Operators in Two-Sorted Frames

• Autorzy: Hartonas Chrysafis , Orłowska Ewa

Identyfikatory

DOI

10.3233/FI-2019-1793

• Języki publikacji: EN

Abstrakty

EN

We study general lattices with normal unary operators for which we prove relational representation and duality results. Similar results have appeared in print, using Urquhart’s lattice representation, by the second author with Vakarelov, Radzikowska and Rewitzky. We base our approach in this article on the Hartonas and Dunn lattice duality, proven by Gehrke and Harding to deliver a canonical lattice extension, and on recent results by the first author on the relational representation of normal lattice operators. We verify that the operators at the representation level (appropriately generated by relations) are the canonical extensions of the lattice operators, in Gehrke and Harding’s sense.

Słowa kluczowe

EN

extension; senses

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2019
• Tom: Vol. 166, nr 1
• Strony: 29--56
• Opis fizyczny: Bibliogr. 31 poz., rys.

Twórcy

autor

Hartonas Chrysafis

• University of Thessaly, Larissa, Greece

autor

Orłowska Ewa

• National Institute of Telecommunications, Szachowa 1, 04-894 Warsaw, Poland

Bibliografia

• [1] Hartonas C. Stone Duality for Lattice Expansions. Oxford Logic Journal of the IGPL, 2018. 26(5):475-504. doi:org/10.1093/jigpal/jzy010.
• [2] Orlowska E, Radzikowska AM, Rewitzky I. Dualities for Structures of Applied Logics. College Publications, 2015. ISBN: 10:9781848901810, 13:978-1848901810.
• [3] Gargov G, Passy S, Tinchev T. Modal environment for Boolean speculations. In: Skordev D (ed.), Mathematical logic and its applications. Springer, Boston, MA, 1987. doi:10.1007/978-1-4613-0897-3)_17.
• [4] Orlowska E, Vakarelov D. Lattice-based modal algebras and modal logics. In: Hájek P, Valdés-Villanueva L, Westerståhl D (eds.), Logic, Methodology and Philosophy of Science, Proceedings of the Twelfth International Congress (713 August 2003, Oviedo, Spain). Kings College Publications, 2005 pp. 147-170.
• [5] Urquhart A. A Topological Representation of Lattices. Algebra Universalis, 1978. 8(1):45-58. doi:10.1007/BF02485369.
• [6] Hartonas C, Dunn JM. Stone Duality for Lattices. Algebra Universalis, 1997.37(3):391-401. doi:10.1007/s000120050024.
• [7] Gehrke M, Harding J. Bounded Lattice Expansions. Journal of Algebra, 2001. 238(1):345-371. URL https://doi.org/10.1006/jabr.2000.8622.
• [8] Craig A, Haviar M. Reconciliation of approaches to the construction of canonical extensions of bounded lattices. Mathematica Slovaka, 2014. 64(6):1335-1356. doi:10.2478/s12175-014-0278-7.
• [9] Moshier MA, Jipsen P. Topological duality and lattice expansions, I: A topological construction of canonical extensions. Algebra universalis, 2014. 71(2):109-126. doi:10.1007/s00012-014-0267-2.
• [10] Moshier MA, Jipsen P. Topological duality and lattice expansions, II: Lattice expansions with quasioperators. Algebra universalis, 2014. 71(3):221-234. doi:10.1007/s00012-014-0275-2.
• [11] Priestley H. Representation of Distributive Lattices by means of Ordered Stone Spaces. Bull. Lond. Math. Soc., 1970. 2:186-190. URL https://doi.org/10.1112/blms/2.2.186.
• [12] Allwein G, Hartonas C. Duality for bounded lattices. Technical Report IULG-93-25, Indiana University Logic Group, 1993.
• [13] Craig A, Haviar M, Priestley H. A Fresh Perspective on Canonical Extensions for Bounded Lattices. Applied Categorical Structures, 2013. 21(6):725-749. doi:10.1007/s10485-012-9287-2.
• [14] Ploščica M. A natural representation of bounded lattices. Tatra Mountains Math. Publ., 1995. 5:75-88.
• [15] Clark DM, Davey BA. Natural Dualities for the Working Algebraist. CUP, 1998. ISBN: 10:0521454158, 13:978-0521454155.
• [16] Hartung G. A topological representation for lattices. Algebra Universalis, 1992. 29:273-299.
• [17] Ganter B, Wille R. Formal Concept Analysis: Mathematical Foundations. Springer, 1999. doi:10.1007/978-3-642-59830-2.
• [18] Hartonas C, Dunn JM. Duality Theorems for Partial Orders, Semilattices, Galois Connections and Lattices. Technical Report IULG-93-26, Indiana University Logic Group, 1993.
• [19] Goldblatt R. Semantic Analysis of Orthologic. Journal of Philosophical Logic, 1974. 3(1-2):19-35. doi:10.1007/BF00652069.
• [20] Jónsson B, Tarski A. Boolean Algebras with Operators I. American Journal of Mathematics, 1951. 73(4):891-939. doi:10.2307/2372123.
• [21] Jónsson B, Tarski A. Boolean Algebras with Operators II. American Journal of Mathematics, 1952. 74(1):127-162. doi:10.2307/2372074.
• [22] Craig A. Canonical Extensions of Bounded Lattices and Natural Duality for Default Bilattices. Ph.D. thesis, University of Oxford, 2012.
• [23] Craig A, Gouveia MJ, Haviar M. TiRS graphs and TiRS frames: a new setting for duals of canonical extensions. Algerba Universalis, 2015. 74(1-2):123-138. doi:10.1007/s00012-015-0335-2.
• [24] Gouveia MJ, Priestley HA. Canonical Extensions and Profinite Completions of Semilattices and Lattices. Order, 2014. 31(2):189-216. doi:10.1007/s11083-013-9296-2.
• [25] Hartonas C. Duality for Lattice-Ordered Algebras and for Normal Algebraizable Logics. Studia Logica, 1997. 58(3):403-450. doi:10.1023/A:1004982417404.
• [26] Allwein G, Dunn JM. Kripke Models for Linear Logic. The Journal of Symbolic Logic, 1993. 58(2):514-545. doi:10.2307/2275217.
• [27] Düntsch I, Orłowska E, Radzikowska A, Vakarelov D. Relational representation theorems for some lattice-based structures. Journal on Relational Methods in Computer Science, 2004. 1:132-160.
• [28] Dzik W, Orlowska E, van Alten C. Relational Representation Theorems for General Lattices with Negations. In: Schmidt RA (ed.), Relations and Kleene Algebra in Computer Science. Springer Berlin Heidelberg, Berlin, Heidelberg, 2006 pp. 162-176. doi:10.1007/11828563_11.
• [29] Hartonas C. Order-Dual Relational Semantics for Non-Distributive Propositional Logics. Oxford Logic Journal of the IGPL, 2017. 25(2):145-182. URL https://doi.org/10.1093/jigpal/jzw057.
• [30] Hartonas C. Order-Dual Relational Semantics for Non-distributive Propositional Logics: A General Framework. Journal of Philosophical Logic, 2018. 47(1):67-94. doi:10.1007/s10992-016-9417-7.
• [31] Hartonas C. Order-Duality, Negation and Lattice Representation. In: Wansing H (ed.), Negation: A Notion in Focus, pp. 27-36. de Gruyter, 1996.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).