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Natural duality, modality, and coalgebra. (English) Zbl 1263.08002
In this paper the author presents a family of topological dualities between classes of algebras with a modal operator and categories of enriched Stone spaces. The classes of algebras considered in the paper are constructed from a semi-primal algebra with a bounded lattice reduct and suitably defining modalities on its powers. Then, appropriately extending the Vietoris functor to each of the enriched Stone spaces, the author also presents a coalgebraic version of the duality.
Several authors have tackled the problem of adding modalities to some classes of algebras with lattice reduct and to obtain topological representations for these classes, e.g., R. Cignoli, S. Lafalce and A. Petrovich [Order 8, No. 3, 299–315 (1991; Zbl 0754.06006)], A. Petrovich [Order 16, No. 1, 1–17 (1999; Zbl 0952.06016)], S. Celani and R. Jansana [Log. J. IGPL 7, No. 6, 683–715 (1999; Zbl 0948.03013)], C. B. Wegener [Stud. Log. 70, No. 3, 339–352 (2002; Zbl 1007.06010)], M. Gehrke, H. Nagahashi and Y. Venema [Ann. Pure Appl. Logic 131, No. 1–3, 65–102 (2005; Zbl 1077.03009)], to mention some of them. These representations usually follow the same strategy used to represent modal algebras (see [G. Sambin and V. Vaccaro, Ann. Pure Appl. Logic 37, No. 3, 249–296 (1988; Zbl 0643.03014)]). That is, given a modal algebra $$A$$, first consider its reduct obtained by dropping its modal operator, then use some topological representation for this reduct (Stone duality, Priestley duality, etc.), then enrich the space that represents the reduct of $$A$$ with a relational structure to represent the modal operator.
The results of this paper are also based on a duality for the reduct without the modal operator. The main difference lies in the fact that, instead of considering the class of modal algebras determined by the properties of its operations, the author’s approach is to consider the class determined in a specific way to construct of modal operators on powers of a semi-primal algebra given a Kripke frame.
This change of perspective has some advantages. First, this approach allows the author to concentrate on the universal properties of the generating algebra that ensure the existence of a duality, rather than on the particularities of the language. Secondly, the fact of having a finite semi-primal algebra as starting point of the process opens access to the Semi-primal Strong Natural Duality Theorem 3.3.14 in [D. M. Clark and B. A. Davey, Natural dualities for the working algebraist. Cambridge: Cambridge University Press (1998; Zbl 0910.08001)], which is the natural duality on which the author bases his construction.
##### MSC:
 08C20 Natural dualities for classes of algebras 03G25 Other algebras related to logic 06D50 Lattices and duality 18B99 Special categories
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##### References:
 [1] S. Abramsky, Domain theory and the logic of observable properties, University of London, 1987. [2] S. Abramsky, A cook’s tour of the finitary non-well-founded sets, We Will Show Them: Essays in honour of Dov Gabbay (2005) 1-18 (this is basically a paper version of an invited talk at the 1988 British Colloquium on Theoretical Computer Science in Edinburgh). [3] Adámek, J., Introduction to coalgebra, Theory appl. categ., 14, 157-199, (2005) · Zbl 1080.18005 [4] Adámek, J.; Herrlich, H.; Strecker, G.E., Abstract and concrete categories, (1990), John Wiley and Sons, Inc. · Zbl 0695.18001 [5] Blackburn, P.; de Rijke, M.; Venema, Y., Modal logic, (2001), CUP · Zbl 0988.03006 [6] Bou, F.; Esteva, F.; Godo, L., On the minimum many-valued modal logic over a finite residuated lattice, J. logic comput., 21, 739-790, (2011) · Zbl 1252.03040 [7] Burris, S.; Sankappanavar, H.P., A course in universal algebra, (1981), Springer-Verlag · Zbl 0478.08001 [8] Cignoli, R.L.O.; Dubuc, E.J.; Mundici, D., Extending stone duality to multisets and locally finite MV-algebras, J. pure appl. algebra, 189, 37-59, (2004) · Zbl 1055.06004 [9] C. Cîrstea, A. Kurz, D. Pattinson, L. Schröder, Y. Venema, Modal logics are coalgebraic, BCS Int. Acad. Conf. (2008) 128-140. [10] Clark, D.M.; Davey, B.A., Natural dualities for the working algebraist, (1998), CUP · Zbl 0910.08001 [11] Davey, B.A.; Priestley, H.A., Introduction to lattices and order, (2002), CUP · Zbl 1002.06001 [12] Davey, B.A.; Schumann, V.J.; Werner, H., From the subalgebra of the square to the discriminator, Algebra universalis, 28, 500-519, (1991) · Zbl 0745.08002 [13] Engelking, R., General topology, (1989), Heldermann Verlag · Zbl 0684.54001 [14] Esakia, L., Topological Kripke models, Sov. math. dokl., 15, 147-151, (1974) · Zbl 0296.02030 [15] Fitting, M.C., Many-valued modal logics, Fund. inform., 15, 235-254, (1991) · Zbl 0745.03018 [16] Fitting, M.C., Many-valued modal logics II, Fund. inform., 17, 55-73, (1992) · Zbl 0772.03006 [17] Goldblatt, R., Mathematics of modality, CSLI lecture notes, vol. 43, (1993), Stanford · Zbl 0942.03516 [18] Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Academic Publishers · Zbl 0937.03030 [19] Hansoul, G., A duality for Boolean algebras with operators, Algebra universalis, 17, 34-49, (1983) · Zbl 0524.06022 [20] G. Hansoul, B. Teheux, Completeness results for many-valued Łukasiewicz modal systems and relational semantics, preprint, arXiv:math/0612542v1. [21] Hut, P.; van Fraassen, B., Elements of reality: a dialogue, J. conscious. stud., 4, 167-180, (1997) [22] Jacobs, B.; Rutten, J., A tutorial on (co)algebras and (co)induction, Bull. eur. assoc. theor. comput. sci., 62, 229-259, (1997) · Zbl 0880.68070 [23] Johnstone, P.T., Stone spaces, (1986), CUP · Zbl 0586.54001 [24] Jónsson, B.; Tarski, A., Boolean algebras with operators I, Amer. J. math., 73, 891-939, (1951) · Zbl 0045.31505 [25] Keimel, K.; Werner, H., Stone duality for varieties generated by a quasi-primal algebra, Mem. amer. math. soc., 148, 59-85, (1974) · Zbl 0283.08001 [26] Kupke, C.; Kurz, A.; Venema, Y., Stone coalgebras, Theoret. comput. sci., 327, 109-134, (2004), (this is an extended version of Electron. Notes Theor. Comput. Sci. 82 (2003) 170-190) · Zbl 1075.68053 [27] Kurz, A., Coalgebras and their logics, SIGACT news, 37, 57-77, (2006) [28] A. Kurz, J. Velebil, Enriched logical connections, preprint. · Zbl 1298.18007 [29] Maruyama, Y., Algebraic study of lattice-valued logic and lattice-valued modal logic, (), 172-186 [30] Maruyama, Y., Fuzzy topology and łukasiewicz logics from the viewpoint of duality theory, Studia logica, 94, 245-269, (2010) · Zbl 1197.03024 [31] Maruyama, Y., Fundamental results for pointfree convex geometry, Ann. pure appl. logic, 161, 1486-1501, (2010) · Zbl 1232.52002 [32] Maruyama, Y., Dualities for algebras of fitting’s many-valued modal logics, Fund. inform., 106, 273-294, (2011) · Zbl 1259.03090 [33] Y. Maruyama, Categorical duality between two aspects of the notion of space, Master thesis, Kyoto University, 2011. [34] Michael, E., Topologies on spaces of subsets, Trans. amer. math. soc., 71, 152-182, (1951) · Zbl 0043.37902 [35] Niederkorn, P., Natural dualities for varieties of MV-algebras, J. math. anal. appl., 225, 58-73, (2001) · Zbl 0974.06006 [36] Nishida, K., An inquiry into the good, (1990), Yale University Press [37] Pitkethly, J.; Davey, B., Dualisability: unary algebras and beyond, (2005), Springer · Zbl 1085.08001 [38] Pixely, A.F., The ternary discriminator function in universal algebra, Math. ann., 191, 167-180, (1971) · Zbl 0203.31201 [39] H.-E. Porst, W. Tholen, Concrete dualities, Category Theory at Work (1991) 111-136. · Zbl 0761.18001 [40] Priestley, H.A., Ordered topological spaces and the representation of distributive lattices, Proc. lond. math. soc., 24, 507-530, (1972) · Zbl 0323.06011 [41] Priestley, H.A., Varieties of distributive lattices with unary operations I, J. aust. math. soc. ser. A, 63, 165-207, (1997) · Zbl 0907.06008 [42] Stone, M.H., The representation of Boolean algebras, Bull. amer. math. soc., 44, 807-816, (1938) · Zbl 0020.34204 [43] M.R. Talukder, Natural duality theory with applications to Heyting algebras and generalised modal algebras, Ph.D. thesis, La Trobe University, 2002. [44] Teheux, B., A duality for the algebras of a łukasiewicz $$n + 1$$-valued modal system, Studia logica, 87, 13-36, (2007) · Zbl 1127.03050 [45] Turi, D.; Rutten, J., On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces, Math. struct. comput. sci., 8, 481-540, (1998) · Zbl 0917.68140
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