Abstract
Formal concept analysis (FCA) associates a binary relation between a set of objects and a set of properties to a lattice of formal concepts defined through a Galois connection. This relation is called a formal context, and a formal concept is then defined by a pair made of a subset of objects and a subset of properties that are put in mutual correspondence by the connection. Several fuzzy logic approaches have been proposed for inducing fuzzy formal concepts from L-contexts based on antitone L-Galois connections. Besides, a possibility-theoretic reading of FCA which has been recently proposed allows us to consider four derivation powerset operators, namely sufficiency, possibility, necessity and dual sufficiency (rather than one in standard FCA). Classically, fuzzy FCA uses a residuated algebra for maintaining the closure property of the composition of sufficiency operators. In this paper, we enlarge this framework and provide sound minimal requirements of a fuzzy algebra w.r.t. the closure and opening properties of antitone L-Galois connections as well as the closure and opening properties of isotone L-Galois connections. We apply these results to particular compositions of the four derivation operators. We also give some noticeable properties which may be useful for building the corresponding associated lattices.
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References
Bělohlávek R. (1999) Fuzzy Galois connections. Mathematical Logic Quarterly 45: 497–504
Bělohlávek R. (2004) Concept lattices and order in fuzzy logic. Annals of Pure and Applied Logic 128(1–3): 277–298
Bělohlávek R., DeBaets B., Outrata J., Vychodil V. (2010) Computing the lattice of all fixpoints of a fuzzy closure operator. IEEE Transactions on Fuzzy Systems 18(3): 546–557
Bělohlávek, R., & Vychodil, V. (2005) What is a fuzzy concept lattice. In Proceedings CLA’05, Third International Conference on Concept Lattices and their Applications. Olomounc, Czech Republic (pp. 34–45).
Bělohlávek R., Vychodil V. (2008) Graded LinClosure and its role in relational data analysis. LNCS 4923: 139–154
Birkhoff G. (1949) Théorie et applications des treillis. Annales de l’IHP 11(5): 227–240
Boulicaut J.F., Besson J. (2008) Actionability and formal concepts: A data mining perspective. International Conference on Formal Concept Analysis LNCS 4933: 14–31
Burusco A., Fuentes-González R. (1994) The study of the L-fuzzy concept lattice. Mathware & Soft Computing 3: 209–218
Burusco A., Fuentes-González R. (1998) Construction of the L-fuzzy concept lattice. Fuzzy Sets and Systems 97(1): 109–114
Chen X., Li Q. (2007) Construction of rough approximations in fuzzy setting. Fuzzy Sets and Systems 158(23): 2641–2653
Djouadi, Y., Dubois, D., & Prade, H. (2010). Possibility theory and formal concept analysis: Context decomposition and uncertainty handling. In Proceedings IPMU’10, 13th International Conference on Information Processing and Management of Uncertainty, Dortmund, Germany, LNCS-LNAI 6178 (pp. 260–269). Springer.
Djouadi, Y., & Prade, H. (2009) Interval-valued fuzzy formal concept analysis. In Proceedings ISMIS’09, 18th International Symposium on Intelligent Systems, Prague, Czech Republic, LNCS-LNAI 5722 (pp. 592–601). Springer.
Djouadi Y., Prade H. (2010) Interval-valued fuzzy Galois connections: Algebraic requirements and concept lattice construction. Fundamenta Informaticae 99(2): 169–186
Dubois D., Dupin deSaintCyr F., Prade H. (2007) A possibilty-theoretic view of formal concept analysis. Fundamenta Informaticae 75(1–4): 195–213
Dubois D., Prade H. (1984) A theorem on implication functions defined from triangular norms. Stochastica 8: 267–279
Dubois, D., & Prade, H. (Eds.). (2009). Possibility theory and formal concept analysis in information systems. Proceedings of IFSA/EUSFLAT’09 (pp. 1021–1026). Lisbon, Portugal.
Düntsch, I., & Gediga, G. (2003). Approximation operators in qualitative data analysis. In Theory and application of relational structures as knowledge instruments (pp. 214–230).
Fan S.Q., Zhang W.X., Ma J.M. (2006) Fuzzy inference based on fuzzy concept lattice. Fuzzy Sets and Systems 157(24): 3177–3187
Fodor, J. C. (1995). Nilpotent minimum and related connectives for fuzzy logic. Fourth IEEE International Conference on Fuzzy Systems (pp. 2077–2082). Yokohama, Japan
Frascella A., Guido C. (2008) Transporting many-valued sets along many-valued relations. Fuzzy Sets and Systems 159(1): 1–22
Ganter B., Wille R. (1999) Formal concept analysis. Springer, Berlin
Gediga, G., & Düntsch, I. (2002). Modal-style operators in qualitative data analysis. In Proceedings ICDM’02 IEEE International Conference on Data Mining, IEEE Computer Society (pp. 155–162).
Georgescu G., Popescu A. (2004) Non-dual fuzzy connections. Arch. Math. Log 43(8): 1009–1039
Goguen J.A. (1967) L-fuzzy sets. Journal of Mathematical Analysis and Applications 18: 145–174
Kuznetsov S.O. (2004) Machine learning and formal concept analysis. International Conference on Formal Concept Analysis, LNCS 2961: 287–312
Lai H., Zhang D. (2009) Concept lattices of fuzzy contexts: formal concept analysis vs rough set theory. International Journal of Approximate Reasoning 150(5): 695–707
Latiri, C. C., Elloumi, S., Chevallet, J. P., & Jaouay, A. (2003). Extension of fuzzy Galois connection for information retrieval using a fuzzy quantifier. In Proceedings AICCSA’03, ACS/IEEE International Conference on Computer Systems and Applications, Tunis, Tunisia (pp. 81–91).
Lei Y., Luo M. (2009) Rough concept lattices and domains. Annals of Pure and Applied Logic 159(33): 333–340
Liu G.L. (2008) Generalized rough sets over fuzzy lattices. Information Sciences 178(6): 1651–1662
Medina J., Ojeda-Aciego M. (2010) Multi-adjoint t-concept lattices. Information Sciences 180(5): 712–725
Medina J., Ojeda-Aciego M., Ruiz-Calviño J. (2009) Formal concept analysis via multi-adjoint concept lattices. Fuzzy Sets and Systems 160(2): 130–144
Pollandt S. (1997) Fuzzy Begriffe. Springer, Berlin
Shao M.W., Liu M., Zhang W.X. (2007) Set approximations in fuzzy formal concept analysis. Fuzzy Sets and Systems 158(23): 2627–2640
Wang X., Zhang W. (2008) Relations of attribute reduction between object and property oriented concept lattices. Knowledge-Based Systems 21(5): 398–403
Ward M., Dilworth R.P. (1939) Residuated lattices. Transaction on American Mathematical Society 45: 335–354
Wille R. (1982) Restructuring lattice theory: An approach based on hierarchies of concepts. In: Rival I (eds) Ordered sets. Reidel, Dordrecht Boston, pp 445–470
Xie, C., Yi, L., & Du, Y. (2007). An algorithm for fuzzy concept lattices building with application to social navigation. Proceedings of ISKE 07, International Conference on Intelligent Systems and Knowledge Engineering. China.
Yao Y.Y., Chen Y. (2006) Rough set approximations in formal concept analysis. Transactions on Rough Sets V, LNCS 4100: 285–305
Zaki M.J. (2005) Efficient mining for mining closed itemsets and their lattice structure. IEEE Transactions on Knowledge and Data Engineering 17(4): 462–478
Zhang W.X., Ma J.M., Fan S.Q. (2007) Variable threshold concept lattices. Information Sciences 177(22): 4883–4892
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Djouadi, Y., Prade, H. Possibility-theoretic extension of derivation operators in formal concept analysis over fuzzy lattices. Fuzzy Optim Decis Making 10, 287–309 (2011). https://doi.org/10.1007/s10700-011-9106-5
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DOI: https://doi.org/10.1007/s10700-011-9106-5