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Lattice-theoretic contexts and their concept lattices via Galois ideals. (English) Zbl 1395.68258
Summary: This paper introduces a concept of lattice-theoretic contexts as well as their concept lattices. A lattice-theoretic context is a triple \((G, M, I)\) with two complete lattices \(G\), \(M\) and their Galois ideal \(I\). A lattice-theoretic context and its concept lattice are a common generalization of classical FCA, Pócs’s formal fuzzy context, one-sided concept lattices, generalized concept lattices and \(L\)-fuzzy concept lattices (with hedges). When the lattices \(G\), \(M\) are completely distributive, a reduction of the relation \(I\) in the lattice-theoretic context \((G, M, I)\) can be obtained. Related algorithms to construct concept lattices of \(L\)-fuzzy contexts considered as lattice-theoretic contexts are presented. In the case of \(L\) being a completely distributive lattice, we can reduce the number of elements (objects or/and attributes) before computing the whole concept lattice. Then the related algorithm has lower complexity.

MSC:
68T30 Knowledge representation
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06D10 Complete distributivity
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[1] Arveson, W., An Invitation to C^*-Algebra, (1976), Springer-Verlag · Zbl 0344.46123
[2] Bělohlávek, R., Fuzzy Galois connections, Math. Log. Q., 45, 497-504, (1999) · Zbl 0938.03079
[3] Bělohlávek, R., Lattices generated by binary fuzzy relations, Tatra Mt. Math. Publ., 16, 11-19, (1999) · Zbl 0949.06002
[4] Bělohlávek, R., Lattices of fixed points of fuzzy Galois connections, Math. Log. Q., 47, 111-116, (2001) · Zbl 0976.03025
[5] Bělohlávek, R., A note on variable threshold concept lattices: threshold-based operators are reducible to classical concept-forming operators, Inf. Sci., 177, 3186-3191, (2007) · Zbl 1119.06004
[6] Bělohlávek, R.; Vychodil, V., Reducing the size of fuzzy concept lattices by hedges, Proceedings of the Fourteenth IEEE International Conference on Fuzzy Systems, 663-668, (May 2005), USA
[7] Yahia, S. B.; Jaoua, A., Discovering knowledge from fuzzy concept lattice, (Kandel, A.; Last, M.; Bunke, H., Data Mining and Computational Intelligence, (2001), Physica-Verlag), 167-190
[8] Burusco, A.; Fuentes-González, R., The study of the L-fuzzy concept lattice, Mathw. Soft Comput., 1, 209-218, (1994) · Zbl 0827.06004
[9] Burusco, A.; Fuentes-González, R., Construction of the L-fuzzy concept lattice, Fuzzy Sets Syst., 97, 109-114, (1998) · Zbl 0927.06006
[10] Butka, P.; Pócs, J.; Pócsová, J., Representation of fuzzy concept lattices in the framework of classical fca, J. Appl. Math., 2013, 1-7, (2013)
[11] Chen, D. G.; Zhang, W. X.; Yeung, D.; Tsang, E. C.C., Rough approximations on a complete completely distributive lattice with applications to generalized rough sets, Inf. Sci., 176, 1829-1848, (2006) · Zbl 1104.03053
[12] Davey, B. A.; Priestley, H., Introduction to Lattices and Order, (2002), Cambridge University Press Cambridge
[13] Dunstan, F. D.J.; Ingleton, A. W.; Welsh, D. J.A., Supermatroids, Proceedings of the Conference on Combinatorial Mathematics, 72-122, (1972), Math. Institute Oxford
[14] Ganter, B.; Wille, R., Formal Concept Analysis: Mathematical Foundations, (1999), Springer-Verlag
[15] Gediga, G.; Düntsch, I., Modal-style operators in qualitative data analysis, Proceedings of the IEEE International Conference on Data Mining, 155-162, (2002)
[16] Georgescu, G.; Popescu, A., Concept lattices and similarity in non-commutative fuzzy logic, Fundam. Inf., 55, 23-54, (2002) · Zbl 1023.03016
[17] Georgescu, G.; Popescu, A., Non-commutative fuzzy Galois connections, Soft Comput., 7, 458-467, (2003) · Zbl 1024.03025
[18] Georgescu, G.; Popescu, A., Non-dual fuzzy connections, Arch. Math. Logic, 43, 1009-1039, (2004) · Zbl 1060.03042
[19] Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S., Continuous Lattices and Domains, (2003), Cambridge University Press Cambridge
[20] Jaoua, A.; Elloumi, S., Galois connection, formal concepts and Galois lattice in real relations: application in a real classifier, J. Syst. Softw., 60, 149-163, (2002)
[21] Johnstone, P. T., Stone Spaces, (1982), Cambridge University Press Cambridge
[22] Kengue, J. F.D.; Valtchev, P.; Djamegni, C. T., A parallel algorithm for lattice construction, Lect. Notes Comput. Sci., 3403, 249-264, (2005) · Zbl 1078.68771
[23] Konecny, J., Isotone fuzzy Galois connections with hedges, Inf. Sci., 181, 1804-1817, (2011) · Zbl 1226.06001
[24] Krajči, S., Cluster based efficient generation of fuzzy concepts, Neural Netw. World, 13, 521-530, (2003)
[25] Krajči, S., A generalized concept lattice, Logic J. IGPL, 13, 543-550, (2005) · Zbl 1088.06005
[26] Medina, J., Multi-adjoint property-oriented and object-oriented concept lattices, Inf. Sci., 190, 95-106, (2012) · Zbl 1248.68479
[27] Medina, J., Relating attribute reduction in formal, object-oriented and property-oriented concept lattices, Comput. Math. Appl., 64, 1992-2002, (2012) · Zbl 1268.06007
[28] Medina, J.; Ojeda-Aciego, M.; Ruiz-Calviño, J., On multi-adjoint concept lattices: definition and representation theorem, Lecture Notes in Computer Science 4390, 197-209, (2007) · Zbl 1187.68588
[29] Medina, J.; Ojeda-Aciego, M.; Ruiz-Calviño, J., Formal concept analysis via multi-adjoint concept lattices, Fuzzy Sets Syst., 160, 130-144, (2009) · Zbl 1187.68589
[30] Ore, O., Galois connexions, Trans. Am. Math. Soc., 55, 493-513, (1944) · Zbl 0060.06204
[31] Picado, J., The quantale of Galois connections, Algebra Univers., 52, 527-540, (2004) · Zbl 1084.06003
[32] Pócs, J., Note on generating fuzzy concept lattices via Galois connections, Inf. Sci., 185, 128-136, (2012) · Zbl 1239.68071
[33] Popescu, A., A general approach to fuzzy concepts, Math. Log. Q., 50, 265-280, (2004) · Zbl 1059.03015
[34] Raney, G. N., Completely distributive complete lattices, Proc. Am. Math. Soc., 3, 677-680, (1952) · Zbl 0049.30304
[35] Rosenthal, K., Quantales and Their Applications, Pitman Research Notes in Mathematics, 234, (1990), Longman · Zbl 0703.06007
[36] Shmuely, Z., The structure of Galois connections, Pac. J. Math., 54, 209-225, (1974) · Zbl 0275.06003
[37] Wang, G. J., Pointwise topology on completely distributive lattices, Fuzzy Sets Syst., 30, 53-62, (1989) · Zbl 0667.54001
[38] Wang, G. J., Theory of topological molecular lattices, Fuzzy Sets Syst., 47, 351-376, (1992) · Zbl 0783.54032
[39] Wang, X.; Zhang, W. X., Relations of attribute reduction between object and property oriented concept lattices, Knowl. Based Syst., 21, 398-403, (2008)
[40] Wang, X.; Zhang, W. X., Concept lattices of subcontexts of a context, Fundam. Inf., 90, 157-169, (2009) · Zbl 1161.06300
[41] Welsh, D. J.A., Matroid Theory, (1976), Academic Press London
[42] Wille, R., Restructuring lattice theory: an approach based on hierarchies of concepts, (Rival, I., Ordered Sets, Reidel, Dordrecht-Boston, (1982)), 445-470
[43] Yao, Y. Y., A comparative study of formal concept analysis and rough set theory in data analysis, Lect. Notes Artif. Intell., 3066, 59-68, (2004) · Zbl 1103.68123
[44] Zhang, W. X.; Ma, J. M.; Fan, S. Q., Variable threshold concept lattices, Inf. Sci., 177, 4883-4892, (2007) · Zbl 1130.06004
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