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Canonical extensions and ultraproducts of polarities. (English) Zbl 06963511

Summary: Jónsson and Tarski’s notion of the perfect extension of a Boolean algebra with operators has evolved into an extensive theory of canonical extensions of lattice-based algebras. After reviewing this evolution we make two contributions. First it is shown that the failure of a variety of algebras to be closed under canonical extensions is witnessed by a particular one of its free algebras. The size of the set of generators of this algebra can be made a function of a collection of varieties and is a kind of Hanf number for canonical closure. Secondly we study the complete lattice of stable subsets of a polarity structure, and show that if a class of polarities is closed under ultraproducts, then its stable set lattices generate a variety that is closed under canonical extensions. This generalises an earlier result of the author about generation of canonically closed varieties of Boolean algebras with operators, which was in turn an abstraction of the result that a first-order definable class of Kripke frames determines a modal logic that is valid in its so-called canonical frames.

MSC:

03G10 Logical aspects of lattices and related structures
06B23 Complete lattices, completions
03C20 Ultraproducts and related constructions
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06D50 Lattices and duality
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[1] Almeida, A., Canonical extensions and relational representations of lattices with negation, Stud. Log., 91, 171-199, (2009) · Zbl 1172.03032
[2] Bayart, A., Quasi-adéquation de la logique modal du second ordre S5 et adéquation de la logique modal du premier ordre S5, Logique et Analyse, 2, 99-121, (1959)
[3] Bell, J.L., Slomson, A.B.: Models and Ultraproducts. North-Holland, Amsterdam (1969) · Zbl 0179.31402
[4] Bezhanishvili, G.; Gehrke, M.; Mines, R.; Morandi, PJ, Profinite completions and canonical extensions of Heyting algebras, Order, 23, 143-161, (2006) · Zbl 1112.06008
[5] Bezhanishvili, G.; Mines, R.; Morandi, PJ, Topo-canonical completions of closure algebras and Heyting algebras, Algebra Universalis, 58, 1-34, (2008) · Zbl 1135.06009
[6] Birkhoff, G.: Lattice Theory, 1st edn. American Mathematical Society, New York (1940) · JFM 66.0100.04
[7] Bruns, G.; Roddy, M., A finitely generated modular ortholattice, Can. Math. Bull., 35, 29-33, (1992) · Zbl 0767.06007
[8] Bulian, J.; Hodkinson, I., Bare canonicity of representable cylindric and polyadic algebras, Anna. Pure Appl. Logic, 164, 884-906, (2013) · Zbl 1321.03077
[9] Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer-Verlag, New York (1981) · Zbl 0478.08001
[10] Burris, S.; Werner, H., Sheaf constructions and their elementary properties, Trans. Am. Math. Soc., 248, 269-309, (1979) · Zbl 0411.03022
[11] Chang, C.C., Keisler, H.J.: Model Theory. North-Holland, Amsterdam (1973) · Zbl 0276.02032
[12] Chernilovskaya, A.; Gehrke, M.; Rooijen, L., Generalized Kripke semantics for the Lambek-Grishin calculus, Logic J. IGPL, 20, 1110-1132, (2012) · Zbl 1301.03025
[13] Conradie, W.; Frittella, S.; Palmigiano, A.; Piazzai, M.; Tzimoulis, A.; Wijnberg, NM; Väänänen, J. (ed.), Categories: How I learned to stop worrying and love two sorts, No. 9803, 145-164, (2016), New York · Zbl 1478.03042
[14] Conradie, W., Palmigiano, A.: Algorithmic correspondence and canonicity for non-distributive logics. arXiv:1603.08515 (2016) · Zbl 1255.03030
[15] Coumans, D., Generalising canonical extension to the categorical setting, Ann. Pure Appl. Logic, 163, 1940-1961, (2012) · Zbl 1263.03062
[16] Coumans, D.; Gehrke, M.; Rooijen, L., Relational semantics for full linear logic, J. Appl. Logic, 12, 50-66, (2014) · Zbl 1335.03063
[17] Craig, A.: Canonical extensions of bounded lattices and natural duality for default bilattices. Ph.D. thesis, University of Oxford (2012)
[18] Craig, A.; Haviar, M., Reconciliation of approaches to the construction of canonical extensions of bounded lattices, Math. Slovaca, 64, 1335-1356, (2014) · Zbl 1349.06010
[19] Craig, APK; Haviar, M.; Priestley, HA, A fresh perspective on canonical extensions for bounded lattices, Appl. Categorical Struct., 21, 725-749, (2013) · Zbl 1318.08004
[20] Cresswell, MJ, A Henkin completeness theorem for T, Notre Dame J. Formal Logic, 8, 186-190, (1967) · Zbl 0183.00802
[21] Davey, BA; Haviar, M.; Priestley, HA, Boolean topological distributive lattices and canonical extensions, Appl. Categorical Struct., 15, 225-241, (2007) · Zbl 1122.06004
[22] Davey, BA; Priestley, H., A topological approach to canonical extensions in finitely generated varieties of lattice-based algebras, Topol. Appl., 158, 1724-1731, (2011) · Zbl 1231.06010
[23] Davey, BA; Priestley, H., Canonical extensions and discrete dualities for finitely generated varieties of lattice-based algebras, Studia Logica, 100, 137-161, (2012) · Zbl 1258.08005
[24] Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990) · Zbl 0701.06001
[25] Dunn, JM; Gehrke, M.; Palmigiano, A., Canonical extensions and relational completeness of some substructural logics, J. Symbol. Logic, 70, 713-740, (2005) · Zbl 1101.03021
[26] Fine, K., Logics containing K4. Part I, J. Symbol. Logic, 39, 31-42, (1974) · Zbl 0287.02010
[27] Fine, Kit, Some Connections Between Elementary and Modal Logic, 15-31, (1975) · Zbl 0316.02021
[28] Gehrke, M., Generalized Kripke frames, Studia Logica, 84, 241-275, (2006) · Zbl 1115.03013
[29] Gehrke, M.; Bezhanishvili, G. (ed.), Canonical extensions, Esakia spaces, and universal models, No. 4, 9-41, (2014), New York · Zbl 1350.03050
[30] Gehrke, M.; Gool, SJ, Distributive envelopes and topological duality for lattices via canonical extensions, Order, 31, 435-461, (2014) · Zbl 1309.06004
[31] Gehrke, M.; Harding, J., Bounded lattice expansions, J. Algebra, 239, 345-371, (2001) · Zbl 0988.06003
[32] Gehrke, M.; Harding, J.; Venema, Y., MacNeille completions and canonical extensions, Trans. Am. Math. Soc., 358, 573-590, (2006) · Zbl 1083.06009
[33] Gehrke, M.; Jansana, R.; Palmigiano, A., Canonical extensions for congruential logics with the deduction theorem, Ann. Pure Appl. Logic, 161, 1502-1519, (2010) · Zbl 1238.03051
[34] Gehrke, M.; Jansana, R.; Palmigiano, A., \(Δ _1\)-completions of a poset, Order, 30, 39-64, (2013) · Zbl 1317.06002
[35] Gehrke, M.; Jónsson, B., Bounded distributive lattices with operators, Math. Japonica, 40, 207-215, (1994) · Zbl 0855.06009
[36] Gehrke, M.; Jónsson, B., Monotone bounded distributive lattice expansions, Math. Japonica, 52, 197-213, (2000) · Zbl 0972.06005
[37] Gehrke, M.; Jónsson, B., Bounded distributive lattice expansions, Math. Scand., 94, 13-45, (2004) · Zbl 1077.06008
[38] Gehrke, M.; Nagahashi, H.; Venema, Y., A Sahlqvist theorem for distributive modal logic, Anna. Pure Appl. Logic, 131, 65-102, (2005) · Zbl 1077.03009
[39] Gehrke, M.; Priestley, HA, Canonical extensions of double quasioperator algebras: An algebraic perspective on duality for certain algebras with binary operations, J. Pure Appl. Algebra, 209, 269-290, (2007) · Zbl 1110.06015
[40] Gehrke, M.; Priestley, HA, Duality for double quasioperator algebras via their canonical extensions, Studia Logica, 86, 31-68, (2007) · Zbl 1127.06009
[41] Gehrke, M.; Priestley, HA, Canonical extensions and completions of posets and lattices, Rep. Math. Logic, 43, 133-152, (2008) · Zbl 1147.06005
[42] Gehrke, M.; Vosmaer, J.; Bezhanishvili, N. (ed.), A view of canonical extension, No. 6618, 77-100, (2009), New York · Zbl 1341.03095
[43] Gehrke, M.; Vosmaer, J., Canonical extensions and canonicity via dcpo presentations, Theor. Comput. Sci., 412, 2714-2723, (2011) · Zbl 1231.06013
[44] Goldblatt, Robert Ian, Metamathematics of modal logic, Bulletin of the Australian Mathematical Society, 10, 479, (1974) · Zbl 0273.02016
[45] Goldblatt, R., Varieties of complex algebras, Ann. Pure Appl. Logic, 44, 173-242, (1989) · Zbl 0722.08005
[46] Goldblatt, R.; Andréka, H. (ed.); Monk, J. (ed.); Németi, I. (ed.), On closure under canonical embedding algebras, No. 54, 217-229, (1991), Amsterdam
[47] Goldblatt, R.: Logics of Time and Computation, second edn. CSLI Lecture Notes No. 7. CSLI Publications, Stanford University (1992)
[48] Goldblatt, R.: Mathematics of Modality. CSLI Lecture Notes No. 43. CSLI Publications, Stanford University (1993)
[49] Goldblatt, R., Elementary generation and canonicity for varieties of Boolean algebras with operators, Algebra Universalis, 34, 551-607, (1995) · Zbl 0854.06020
[50] Goldblatt, R.: Fine’s theorem on first-order complete modal logics. arXiv:1604.02196 (2016)
[51] Goldblatt, R., Hodkinson, I.: The McKinsey-Lemmon logic is barely canonical. Aust. J. Logic 5, 1-19 (2007). https://ojs.victoria.ac.nz/ajl/article/view/1783 · Zbl 1168.03320
[52] Goldblatt, R.; Hodkinson, I.; Venema, Y., On canonical modal logics that are not elementarily determined, Logique et Analyse, 181, 77-101, (2003) · Zbl 1060.03037
[53] Goldblatt, R.; Hodkinson, I.; Venema, Y., Erdős graphs resolve Fine’s canonicity problem, Bull. Symb. Logic, 10, 186-208, (2004) · Zbl 1060.03038
[54] González, LJ; Jansana, R., A topological duality for posets, Algebra Universalis, 76, 455-478, (2016) · Zbl 1397.06006
[55] Gool, SJ, Duality and canonical extensions for stably compact spaces, Ann. Pure Appl. Logic, 159, 341-359, (2012) · Zbl 1233.54021
[56] Gouveia, MJ; Priestley, HA, Profinite completions and canonical extension of semilttice reducts of distributive lattices, Houston J. Math., 39, 1117-1136, (2013) · Zbl 1312.06004
[57] Gouveia, MJ; Priestley, HA, Canonical extensions and profinite completions of semilattices and lattices, Order, 31, 189-216, (2014) · Zbl 1301.06022
[58] Harding, J., Canonical completions of lattices and ortholattices, Tatra Mt. Math. Publ., 15, 89-96, (1998) · Zbl 0939.06004
[59] Harding, J., The free orthomodular lattice on countably many generators is a subalgebra of the free orthomodular lattice on three generators, Algebra Universalis, 48, 171-182, (2002) · Zbl 1061.06019
[60] Harding, J., On profinite completions and canonical extensions, Algebra Universalis, 55, 293-296, (2006) · Zbl 1134.06004
[61] Hartung, G., A topological representation of lattices, Algebra Universalis, 29, 273-299, (1992) · Zbl 0790.06005
[62] Haviar, M.; Priestley, HA, Canonical extensions of Stone and double Stone algebras: the natural way, Math. Slovaca, 56, 53-78, (2006) · Zbl 1164.06317
[63] Henkin, L., The completeness of the first-order functional calculus, J. Symb. Logic, 14, 159-166, (1949) · Zbl 0034.00602
[64] Henkin, L., Monk, J.D., Tarski, A.: Cylindric Algebras I. North-Holland, Amsterdam (1971) · Zbl 0214.01302
[65] Herrmann, C., A finitely generated modular ortholattice, Can. Math. Bull., 24, 241-243, (1981) · Zbl 0457.06006
[66] Hodkinson, I.; Venema, Y., Canonical varieties with no canonical axiomatisation, Trans. Am. Math. Soc., 357, 4579-4605, (2005) · Zbl 1081.03062
[67] Hughes, GE; Cresswell, MJ, K1.1 is not canonical. Bulletin of the Section of Logic, Pol. Acad. Sci., 11, 109-112, (1982) · Zbl 0525.03003
[68] Jónsson, B.: Canonical extensions of bounded distributive lattice expansions. Abstract and notes for an invited talk at the International Conference on Order, Algebra and Logics, Vanderbilt University, June 2007. www.math.vanderbilt.edu/ oal2007/submissions/Jonsson.pdf
[69] Jónsson, B.: The role of universal algebra and lattice theory. Abstract of an invited talk at the 1992 New Zealand Mathematics Colloquium, Victoria University of Wellington
[70] Jónsson, B., A survey of Boolean algebras with operators, No. 389, 239-286, (1993), Norwell · Zbl 0811.06012
[71] Jónsson, B., On the canonicity of Sahlqvist identities, Studia Logica, 53, 473-491, (1994) · Zbl 0810.03050
[72] Jónsson, B., The preservation theorem for canonical extensions of Boolean algebras with operators. In: K.A. Baker, R. Wille (eds.) Lattice Theory and its Applications, No. 23, 121-130, (1995), Berlin
[73] Jónsson, B.; Tarski, A., Boolean algebras with operators, Bull. Am. Math. Soc., 54, 79-80, (1948)
[74] Jónsson, B.; Tarski, A., Boolean algebras with operators, part I, Am. J. Math., 73, 891-939, (1951) · Zbl 0045.31505
[75] Jónsson, B.; Tarski, A., Boolean algebras with operators, part II, Am. J. Math., 74, 127-162, (1952) · Zbl 0045.31601
[76] Kalmbach, G.: Orthomodular Lattices. Academic Press, Cambridge (1983) · Zbl 0512.06011
[77] Kikot, S., A dichotomy for some elementarily generated modal logics, Studia Logica, 103, 1063-1093, (2015) · Zbl 1373.03023
[78] Kotas, J., An axiom system for the modular logic, Studia Logica, 21, 17-38, (1967) · Zbl 0333.02023
[79] Lemmon, E.J.: An Introduction to Modal Logic, American Philosophical Quarterly Monograph Series, vol. 11. Basil Blackwell, Oxford (1977). (Written in 1966 in collaboration with Dana Scott. Edited by Krister Segerberg) · Zbl 0388.03006
[80] Lemmon, E.J., Scott, D.: Intensional logic (1966). Preliminary draft of initial chapters by E. J. Lemmon, Stanford University (later published as [79])
[81] MacNeille, HM, Partially ordered sets, Trans. Am. Math. Soc., 42, 416-460, (1937) · JFM 63.0833.04
[82] Makinson, DC, On some completeness theorems in modal logic, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 12, 379-384, (1966) · Zbl 0295.02014
[83] McKenzie, R.N., McNulty, G.F., Taylor, W.F.: Algebras, Lattices, Varieties, vol. 1. Wadsworth & Brooks/Cole, Belmont (1987) · Zbl 0611.08001
[84] McKinsey, JCC; Tarski, A., The algebra of topology, Ann. Math., 45, 141-191, (1944) · Zbl 0060.06206
[85] Morton, W., Canonical extensions of posets, Algebra Universalis, 72, 167-200, (2014) · Zbl 1302.06002
[86] Moshier, MA; Jipsen, P., Topological duality and lattice expansions, I: A topological construction of canonical extensions, Algebra Universalis, 71, 109-126, (2014) · Zbl 1307.06002
[87] Moshier, MA; Jipsen, P., Topological duality and lattice expansions, II: Lattice expansions with quasioperators, Algebra Universalis, 71, 221-234, (2014) · Zbl 1307.06003
[88] Priestley, HA, Representations of distributive lattices by means of ordered Stone spaces, Bull. Lond. Math. Soc., 2, 186-190, (1970) · Zbl 0201.01802
[89] Rijke, M.; Venema, Y., Sahlqvist’s theorem for Boolean algebras with operators with an application to cylindric algebras, Studia Logica, 54, 61-78, (1995) · Zbl 0823.03037
[90] Sahlqvist, Henrik, Completeness and Correspondence in the First and Second Order Semantics for Modal Logic, 110-143, (1975) · Zbl 0319.02018
[91] Segerberg, K., Decidability of S4.1, Theoria, 34, 7-20, (1968)
[92] Segerberg, K.: An Essay in Classical Modal Logic, Filosofiska Studier, vol. 13. Uppsala Universitet (1971) · Zbl 0311.02028
[93] Smoryński, C., Fixed point algebras, Bull. Am. Math. Soc., 6, 317-356, (1982) · Zbl 0544.03032
[94] Stone, MH, The theory of representations for Boolean algebras, Trans. Am. Math. Soc., 40, 37-111, (1936) · JFM 62.0033.04
[95] Suzuki, T., Canonicity results of substructural and lattice-based logics, Rev. Symb. Logic, 4, 1-42, (2011) · Zbl 1229.03023
[96] Suzuki, T., On canonicity of poset expansions, Algebra Universalis, 66, 243-276, (2011) · Zbl 1230.03094
[97] Thomason, SK, Semantic analysis of tense logic, J. Symb. Logic, 37, 150-158, (1972) · Zbl 0238.02027
[98] Urquhart, A., A topological representation theory for lattices, Algebra Universalis, 8, 45-58, (1978) · Zbl 0382.06010
[99] Vosmaer, J.: Logic, algebra and topology: Investigations into canonical extensions, duality theory and point-free topology. Ph.D. thesis, Institute for Logic, Language and Computation, Universiteit van Amsterdam (2010). ILLC Dissertation Series DS-2010-10
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