Lattices with interior and closure operators and abstract approximation spaces. (English) Zbl 1248.06005
Peters, James F. (ed.) et al., Transactions on Rough Sets X. Berlin: Springer (ISBN 978-3-642-03280-6/pbk). Lecture Notes in Computer Science 5656. Journal Subline, 67-116 (2009).
Summary: The non-equational notion of abstract approximation space for roughness theory is introduced, and its relationship with the equational definition of lattice with Tarski interior and closure operations is studied. Their categorical isomorphism is proved, and the role of the Tarski interior and closure with an algebraic semantic of an S4-like model of modal logic is widely investigated.
A hierarchy of three particular models of this approach to roughness based on a concrete universe is described, listed from the stronger model to the weaker one: (1) the partition spaces, (2) the topological spaces by open basis, and (3) the covering spaces.
For the entire collection see [Zbl 1169.68304].
A hierarchy of three particular models of this approach to roughness based on a concrete universe is described, listed from the stronger model to the weaker one: (1) the partition spaces, (2) the topological spaces by open basis, and (3) the covering spaces.
For the entire collection see [Zbl 1169.68304].
MSC:
| 06B75 | Generalizations of lattices |
| 03B45 | Modal logic (including the logic of norms) |
| 03G25 | Other algebras related to logic |
| 68T37 | Reasoning under uncertainty in the context of artificial intelligence |
References:
| [1] | Bialynicki-Birula, A.: Remarks on quasi–Boolean algebras. Bull. Acad. Pol. Sci. Cl III 5, 615–619 (1957) · Zbl 0086.01002 |
| [2] | Bialynicki-Birula, A., Rasiowa, H.: On the representation of quasi–Boolean algebras. Bull. Acad. Pol. Sci. Cl III 5, 259–261 (1957) · Zbl 0082.01403 |
| [3] | Birkhoff, G.: Lattice theory, 3rd edn. American Mathematical Society Colloquium Publication, vol. XXV. American Mathematical Society, Providence (1967); (first edition 1940, second (revisited) edition (1948) · Zbl 0033.10103 |
| [4] | Cattaneo, G.: Generalized rough sets (preclusivity fuzzy-intuitionistic BZ lattices). Studia Logica 58, 47–77 (1997) · Zbl 0864.03040 · doi:10.1023/A:1004939914902 |
| [5] | Cattaneo, G.: Abstract approximation spaces for rough theories. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery 1, pp. 59–98. Physica–Verlag, Heidelberg (1998) · Zbl 0927.68087 |
| [6] | Cattaneo, G., Ciucci, D.: About the Lattice Structure of Preclusive Rough Sets. In: IEEE International Conference on Fuzzy Systems, Budapest, July 25-28 (2004) · Zbl 1103.68835 |
| [7] | Cattaneo, G., Ciucci, D.: Algebraic structures for rough sets. In: Dubois, D., Grzymala-Busse, J.W., Inuiguchi, M., Polkowski, L. (eds.) Transactions on Rough Sets II. LNCS, vol. 3135, pp. 218–264. Springer, Heidelberg (2004) · doi:10.1007/978-3-540-27778-1_12 |
| [8] | Cattaneo, G., Ciucci, D.: Investigation about Time Monotonicity of Similarity and Preclusive Rough Approximations in Incomplete Information Systems. In: Tsumoto, S., Słowiński, R., Komorowski, J., Grzymała-Busse, J.W. (eds.) RSCTC 2004. LNCS (LNAI), vol. 3066, pp. 38–48. Springer, Heidelberg (2004) · doi:10.1007/978-3-540-25929-9_4 |
| [9] | Cattaneo, G., Ciucci, D.: Basic intuitionistic principles in fuzzy set theories and its extensions (a terminological debate on Atanassov IFS). Fuzzy sets and Systems 157, 3198–3219 (2006) · Zbl 1112.03050 · doi:10.1016/j.fss.2006.06.003 |
| [10] | Cattaneo, G., Ciucci, D.: Some methodological remarks about categorical equivalence in the abstract approach to roughness. Part I. In: Wang, G.-Y., Peters, J.F., Skowron, A., Yao, Y. (eds.) RSKT 2006. LNCS (LNAI), vol. 4062, pp. 277–283. Springer, Heidelberg (2006) · Zbl 1196.03071 · doi:10.1007/11795131_40 |
| [11] | Cattaneo, G., Ciucci, D.: Some methodological remarks about categorical equivalence in the abstract approach to roughness. Part II. In: Wang, G.-Y., Peters, J.F., Skowron, A., Yao, Y. (eds.) RSKT 2006. LNCS (LNAI), vol. 4062, pp. 284–289. Springer, Heidelberg (2006) · Zbl 1196.03070 · doi:10.1007/11795131_41 |
| [12] | Cattaneo, G., Ciucci, D.: A hierarchical lattice closure approach to abstract approximation spaces. In: Wang, G., Li, T., Grzymala-Busse, J.W., Miao, D., Skowron, A., Yao, Y. (eds.) RSKT 2008. LNCS (LNAI), vol. 5009, pp. 363–370. Springer, Heidelberg (2008) · doi:10.1007/978-3-540-79721-0_51 |
| [13] | Chellas, B.F.: Modal logic, an introduction. Cambridge University Press, Cambridge (1988) · Zbl 0431.03009 |
| [14] | Cignoli, R.: Boolean elements in Łukasiewicz algebras. I. Proceedings of the Japan Academy 41, 670–675 (1965) · Zbl 0168.00601 · doi:10.3792/pja/1195522292 |
| [15] | Cignoli, R.: Injective de Morgan and Kleene algebras. Proceedings of the American Mathematical Society 47, 269–278 (1975) · Zbl 0301.06009 · doi:10.1090/S0002-9939-1975-0357259-4 |
| [16] | Ciucci, D.: A unifying abstract approach for rough models. In: Wang, G., Li, T., Grzymala-Busse, J.W., Miao, D., Skowron, A., Yao, Y. (eds.) RSKT 2008. LNCS (LNAI), vol. 5009, pp. 371–378. Springer, Heidelberg (2008) · doi:10.1007/978-3-540-79721-0_52 |
| [17] | Cattaneo, G., Marino, G.: Non-usual orthocomplementations on partially ordered sets and fuzziness. Fuzzy Sets and Systems 25, 107–123 (1988) · Zbl 0631.06005 · doi:10.1016/0165-0114(88)90104-2 |
| [18] | Cattaneo, G., Nisticò, G.: Brouwer-Zadeh posets and three valued Łukasiewicz posets. Fuzzy Sets and Systems 33, 165–190 (1989) · Zbl 0682.03036 · doi:10.1016/0165-0114(89)90239-X |
| [19] | Düntsch, I., Gediga, G.: Approximation operators in qualitative data analysis. In: de Swart, H., Orłowska, E., Schmidt, G., Roubens, M. (eds.) Theory and Applications of Relational Structures as Knowledge Instruments. LNCS, vol. 2929, pp. 214–230. Springer, Heidelberg (2003) · Zbl 1203.68193 · doi:10.1007/978-3-540-24615-2_10 |
| [20] | Düntsch, I., Orlowska, E.: Beyond modalities: Sufficiency and mixed algebras. In: Orlowska, E., Szalas, A. (eds.) Relational Methods for Computer Science Applications, pp. 277–299. Physica–Verlag, Heidelberg (2001) |
| [21] | Dunn, J.M.: Relevance logic and entailment. In: Gabbay, D., Guenther, F. (eds.) Handbook of Philosophical Logic, vol. 3, pp. 117–224. Kluwer, Dordrecht (1986) · Zbl 0875.03051 · doi:10.1007/978-94-009-5203-4_3 |
| [22] | Everett, C.J.: Closure operators and Galois theory in lattices. Transaction of the American Mathematical Society 55, 514–525 (1944) · Zbl 0060.06205 · doi:10.1090/S0002-9947-1944-0010556-9 |
| [23] | Goldblatt, R.: Mathematical modal logic: A view of its evolution. J. Applied Logic 1, 309–392 (2003) · Zbl 1041.03015 · doi:10.1016/S1570-8683(03)00008-9 |
| [24] | Grzymala-Busse, J.W., Grzymala-Busse, W.J.: Handling missing attribute values. In: Maimon, O., Rokach, L. (eds.) The Data Mining and Knowledge Discovery Handbook, pp. 37–57. Springer, Heidelberg (2005) · Zbl 1156.68583 · doi:10.1007/0-387-25465-X_3 |
| [25] | Grzymala-Busse, J.W., Rzasa, W.: Local and global approximations for incomplete data. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets VIII. LNCS, vol. 5084, pp. 21–34. Springer, Heidelberg (2008) · Zbl 1162.68689 · doi:10.1007/978-3-540-85064-9_2 |
| [26] | Grzymala-Busse, J.W.: Data with missing attribute values: Generalization of indiscernibility relation and rule induction. In: Peters, J.F., Skowron, A., Grzymała-Busse, J.W., Kostek, B.z., Świniarski, R.W., Szczuka, M.S. (eds.) Transactions on Rough Sets I. LNCS, vol. 3100, pp. 78–95. Springer, Heidelberg (2004) · Zbl 1104.68759 · doi:10.1007/978-3-540-27794-1_3 |
| [27] | Hardegree, G.M.: The conditional in abstract and concrete quantum logic. In: Hooker, C.A. (ed.) Logico–Algebraic approach to quantum mechanics. II, pp. 49–108. D. Reidel, Dordrecht (1979) · doi:10.1007/978-94-009-9351-8_4 |
| [28] | Hardegree, G.M.: Material implication in orthomodular (and Boolean) lattices. Notre Dame Journal of Modal Logic 22, 163–182 (1981) · Zbl 0438.03060 · doi:10.1305/ndjfl/1093883401 |
| [29] | Hughes, G.E., Cresswell, M.J.: A companion to modal logic. Methuen, London (1984) · Zbl 0625.03005 |
| [30] | Järvinen, J.: Pawlak’s information systems in terms of Galois conections and functional dependencies. Fundamenta Informaticae 75, 315–330 (2007) · Zbl 1108.68115 |
| [31] | Järvinen, J., Kondo, M., Kortelainen, J.: Modal-like operators in Boolean lattices, Galois connections and fixed points. Fundamenta Informaticae 76, 129–145 (2007) · Zbl 1117.03069 |
| [32] | Kalman, J.A.: Lattices with involution. Transactions of the American Mathematica Society 87, 485–491 (1958) · Zbl 0228.06003 · doi:10.1090/S0002-9947-1958-0095135-X |
| [33] | Kripke, S.A.: Semantical analysis of modal logic I. Normal modes propositional calculi. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 9, 67–96 (1963) · Zbl 0118.01305 · doi:10.1002/malq.19630090502 |
| [34] | Kryszkiewicz, M.: Rough set approach to incomplete information systems. Information Sciences 112, 39–49 (1998) · Zbl 0951.68548 · doi:10.1016/S0020-0255(98)10019-1 |
| [35] | Mac Lane, S.: Categories for the working mathematicians. Graduate Text in Mathematics, vol. 5. Springer, Heidelberg (1971) · Zbl 0232.18001 · doi:10.1007/978-1-4612-9839-7 |
| [36] | Latkowski, R.: Flexible indiscernibility relations for missing attribute values. Fundamenta informaticae 67, 131–147 (2005) · Zbl 1096.68149 |
| [37] | Moisil, G.C.: Recherches sur l’algebres de la logiques. Annales Sc. Univ. Jassy 22, 1–117 (1935) |
| [38] | Monteiro, A., Ribeiro, H.: L’operation de fermeture et ses invariants dans les systemes partiellement ordennes. Portugaliae Mathematica 3, 171–184 (1942) · JFM 68.0508.01 |
| [39] | Ore, O.: Combinations of closure relations. Annals of Mathematics 44, 514–533 (1942) · Zbl 0060.06203 · doi:10.2307/1968978 |
| [40] | Ore, O.: Galois connexions. Transactions of the American Mathematical Society 55, 493–513 (1944) · Zbl 0060.06204 · doi:10.1090/S0002-9947-1944-0010555-7 |
| [41] | Orlowska, E.: Kripke semantics for knowledge representation logics. Studia Logica 49, 255–272 (1990) · Zbl 0726.03023 · doi:10.1007/BF00935602 |
| [42] | Orlowska, E.: Introduction: What you always wanted to know about rough sets. In: Orlowska, E. (ed.) Incomplete Information: Rough Set Analysis, pp. 1–20. Physica–Verlag, Heidelberg (1998) · doi:10.1007/978-3-7908-1888-8_1 |
| [43] | Pawlak, Z.: Information systems - theoretical foundations. Information Systems 6, 205–218 (1981) · Zbl 0462.68078 · doi:10.1016/0306-4379(81)90023-5 |
| [44] | Pawlak, Z.: Rough sets. Int. J. of Computer and Information Sciences 11, 341–356 (1982) · Zbl 0501.68053 · doi:10.1007/BF01001956 |
| [45] | Pawlak, Z.: Rough sets: A new approach to vagueness. In: Zadeh, L.A., Kacprzyc, J. (eds.) Fuzzy Logic for the Management of Uncertainty, pp. 105–118. J. Wiley and Sons, New York (1992) |
| [46] | Poincaré, H.: Le continu mathématique. Revue de Métaphysique et de Morale I, 26–34 (1893) (reprinted in [47] as Chapter II) |
| [47] | Poincaré, H.: La science et l’hypothèse. Flammarion, Paris (1903); English translation as Science and Hypothesis. Dover, New York (1952) |
| [48] | Polkowski, L.: Rough sets. Mathematical foundations. Physica Verlag, Heidelberg (2002) · Zbl 1040.68114 · doi:10.1007/978-3-7908-1776-8 |
| [49] | Pomykala, J.A.: Approximation operations in approximation space. Bulletin of the Polish Academy of Sciences - Mathematics 35, 653–662 (1987) |
| [50] | Polkowski, L., Skowron, A., Zytkow, J.: Tolerance based rough sets. In: Lin, T.Y., Wildberger, A.M. (eds.) Third International Workshop on Rough Sets and Soft Computing, University of San Jose, California, pp. 55–58 (1994) |
| [51] | Simmons, G.F.: Topology and modern analysis. McGraw-Hill Book Company, Inc., New York (1963) · Zbl 0105.30603 |
| [52] | Skowron, A., Stepaniuk, J.: Tolerance approximation spaces. Fundamenta Informaticae 27, 245–253 (1996) · Zbl 0868.68103 |
| [53] | Stefanowski, J., Tsoukias, A.: Incomplete information tables and rough classification. Computational Intelligence 17, 545–566 (2001) · doi:10.1111/0824-7935.00162 |
| [54] | Słowinski, R., Vanderpooten, D.: Similarity relation as a basis for rough approximations. In: Wang, P.P. (ed.) Advances in Machine Intelligence and Soft-Computing, vol. IV, pp. 17–33. Duke University Press, Durham (1997) |
| [55] | Tarski, A.: Fundamentale Begriffe der Methodologie der deduktiven Wissennschaften. I. Monathshefte fur Mathematik und Physik 37, 361–404 (1930) (English version in [56]) · JFM 56.0046.02 · doi:10.1007/BF01696782 |
| [56] | Tarski, A.: Logic, semantics, metamathematics. Hackett, Indianapolis (1983); Second Edition–First Edition, by Oxford 1956 |
| [57] | van Frassen, B.C.: Formal semantic and logic. Macmillan, New York (1971) |
| [58] | Ward, M.: The closure operator on lattice. Annals of Mathematics 43, 191–196 (1942) · Zbl 0063.08179 · doi:10.2307/1968865 |
| [59] | Wiweger, A.: On topological rough sets. Bullettin of the Polish Academy of Sciences–Mathematics 37, 89–93 (1989) · Zbl 0755.04010 |
| [60] | Yao, Y.Y.: Two views of the theory of rough sets in finite universes. International Journal of Approximate Reasoning 15, 291–317 (1996) · Zbl 0935.03063 · doi:10.1016/S0888-613X(96)00071-0 |
| [61] | Yao, Y.Y.: On generalizing Pawlak approximation operators. In: Polkowski, L., Skowron, A. (eds.) RSCTC 1998. LNCS (LNAI), vol. 1424, pp. 298–307. Springer, Heidelberg (1998) · Zbl 0955.68505 · doi:10.1007/3-540-69115-4_41 |
| [62] | Yao, Y.Y.: Relational interpretations of neighborhood operators and rough set approximation operators. Information Sciences 111, 239–259 (1998) · Zbl 0949.68144 · doi:10.1016/S0020-0255(98)10006-3 |
| [63] | Yao, Y.Y.: A note on definability and approximations. In: Peters, J.F., Skowron, A., Marek, V.W., Orłowska, E., Słowiński, R., Ziarko, W.P. (eds.) Transactions on Rough Sets VII. LNCS, vol. 4400, pp. 274–282. Springer, Heidelberg (2007) · Zbl 1187.68617 · doi:10.1007/978-3-540-71663-1_17 |
| [64] | Yao, Y.Y., Lin, T.Y.: Generalization of rough sets using modal logic. Intelligent Automation and Soft Computing, an International Journal 2, 103–120 (1996) · doi:10.1080/10798587.1996.10750660 |
| [65] | Yao, Y., Li, X., Lin, T., Liu, Q.: Representation and classification of rough set models. In: Conference Proceeding of Third International Workshop on Rough Sets and Soft Computing, San Jose, California, November 10-12, pp. 630–637 (1994) |
| [66] | Yao, Y.Y., Wong, S.K.M., Lin, T.Y.: A review of rough set models. In: Lin, T.Y., Cercone, N. (eds.) Rough Sets and Data Mining: Analysis for Inprecise Data, Boston, pp. 47–75. Kluwer, Dordrecht (1997) · Zbl 0861.68101 · doi:10.1007/978-1-4613-1461-5_3 |
| [67] | Zakowski, W.: Approximations in the space (U,{\(\pi\)}). Demonstration Mathematica XVI, 761–769 (1983) |
| [68] | Zeeman, E.C.: The topology of the brain and visual perception. Topology of 3-manifolds and related topics, pp. 240–256. Prentice-Hall, Englewood Cliffs (1962) · Zbl 1246.92006 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
