zbMATH — the first resource for mathematics

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.

68T30 Knowledge representation
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06D10 Complete distributivity
Full Text: DOI
[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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.