^{1}School of Mathematics and Statistics, Victoria University of Wellington, New Zealand

^{2}Department of Computing, Imperial College London, UK.

Abstract

The tangled closure of a collection of subsets of a topological space is the largest subset in which each member of the collection is dense. This operation models a logical `tangle modality' connective, of significance in finite model theory. Here we study an abstract equational algebraic formulation of the operation which generalises the McKinsey-Tarski theory of closure algebras. We show that any dissectable tangled closure algebra, such as the algebra of subsets of any metric space without isolated points, contains copies of every finite tangled closure algebra. We then exhibit an example of a tangled closure algebra that cannot be embedded into any complete tangled closure algebra, so it has no MacNeille completion and no spatial representation.

Highlights

Dedicated to Bernhard Banaschewski on the occasion of his 90th birthday

[1] Banaschewski, B., Hullensysteme und Erweiterung von Quasi-Ordnungen, Z. Math. Log. Grundlagen Math. 2 (1956), 117-130, [2] Banaschewski, B. and Bruns, G., Categorical characterization of the MacNeille completion, Arch. Math. 18 (1967), 369-377.

[3] Davey, B.A. and Priestley, H.A., "Introduction to Lattices and Order", Cambridge University Press, 1990. [4] Dawar, A. and Otto, M., Modal characterisation theorems over special classes of frames, Ann. Pure Appl. Logic 161 (2009), 1-42. [5] Dummett, M.A.E. and Lemmon, E.J., Modal logics between S4 and S5, Z. Math. Log. Grundlagen Math., 5 (1959), 250-264. [6] Fernandez-Duque, D., Tangled modal logic for spatial reasoning, In Toby Walsh, ed- itor, Proceedings of the Twenty-Second International Joint Conference on Articial Intelligence (IJCAI), AAAI Press/IJCAI (2011), 857-862. [7] Fernandez-Duque, D., Tangled modal logic for topological dynamics, Ann. Pure Appl. Logic 163 (2012), 467-481. [8] Givant, S. and Halmos, P., Introduction to Boolean Algebras", Springer, 2009. [9] Goldblatt, R. and Hodkinson I., The nite model property for logics with the tangle modality, Submitted. [10] Goldblatt, R. and Hodkinson I., Spatial logic of modal mu-calculus and tangled closure operators, arxiv.org/abs/1603.01766, 2016. [11] Goldblatt, R. and Hodkinson I., Spatial logic of tangled closure operators and modal mu-calculus, Ann. Pure Appl. Logic, Available online, Nov. 2016: http://dx.doi.org/10.1016/j.apal.2016.11.006. [12] Goldblatt, R. and Hodkinson, I., The tangled derivative logic of the real line and zerodimensional spaces, In Advances in Modal Logic" Volume 11, Lev Beklemishev, Stephane Demri, and Andras Mate, editors, College Publications, 2016, 342-361. [13] Johnstone P., Elements of the history of locale theory, In Handbook of the History of General Topology" Volume 3, C.E. Aull and R. Lowen, editors, Kluwer Academic Publishers, 2001, 835-851. [14] Jonsson, B. and Tarski, A., Boolean algebras with operators, I, Amer. J. Math. 73 (1951), 891-939. [15] Kuratowski, C., Sur l'operation A de l'Analysis Situs, Fund. Math. 3 (1922), 182- 199. [16] MacNeille, H.M., Partially ordered sets, Trans. Amer. Math. Soc. 42 (1937), 416-460. [17] McKinsey, J.C.C. and Tarski, A., The algebra of topology, Ann. Math. 45 (1944), 141-191.

[18] McKinsey, J.C.C. and Tarski, A., On closed elements in closure algebras, Ann. Math. 47 (1946), 122-162. [19] Monk, J.D., Completions of Boolean algebras with operators, Math. Nachr. 46 (1970), 47-55. [20] Rasiowa, H., Algebraic treatment of the functional calculi of Heyting and Lewis, Fund. Math. 38 (1951), 99-126. [21] Rasiowa, H. and Sikorski, R., The Mathematics of Metamathematics", PWN{ Polish Scientic Publishers, Warsaw, 1963. [22] Sikorski, R., Boolean Algebras", Springer-Verlag, Berlin, 1964. [23] Tarski, A., Der Aussagenkalkul und die Topologie, Fund. Math. 31 (1938), 103-134. English translation by J.H. Woodger as Sentential calculus and topology in [24], 421-454. [24] Tarski, A., Logic, Semantics, Metamathematics: Papers from 1923 to 1938", Ox- ford University Press, 1956. Translated into English and edited by J.H. Woodger. [25] Theunissen, M. and Venema, Y., MacNeille completions of lattice expansions, Al- gebra Universalis 57 (2007), 143-193. [26] Van Benthem, J.F.A.K., Modal correspondence theory", PhD Thesis, University of Amsterdam, 1976. [27] Van Benthem, J.F.A.K., Modal Logic and Classical Logic", Bibliopolis, Naples, 1983.