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On the Dedekind-MacNeille completion and formal concept analysis based on multilattices. (English) Zbl 1386.06003

Summary: The Dedekind-MacNeille completion of a poset \(P\) can be seen as the least complete lattice containing \(P\). In this work, we analyze some results concerning the use of this completion within the framework of formal concept analysis in terms of the poset of concepts associated with a Galois connection between posets. Specifically, we show an interesting property of the Dedekind-MacNeille completion, in that the completion of the concept poset of a Galois connection between posets coincides with the concept lattice of the Galois connection extended to the corresponding completions. Moreover, we study the specific case when \(P\) has multilattice structure and state and prove the corresponding representation theorem.

MSC:

06B23 Complete lattices, completions
06A15 Galois correspondences, closure operators (in relation to ordered sets)
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[1] Antoni, L.; Krajci, S.; Kridlo, O.; Macek, B.; Pisková, L., On heterogeneous formal contexts, Fuzzy Sets Syst., 234, 22-33, (2014) · Zbl 1315.68232
[2] Bělohlávek, R., Concept lattices and order in fuzzy logic, Ann. Pure Appl. Logic, 128, 277-298, (2004) · Zbl 1060.03040
[3] Benado, M., LES ensembles partiellement ordonnés et le théoréme de raffinement de Schreier I, Czechoslov. Math. J., 4, 2, 105-129, (1954) · Zbl 0056.04602
[4] Benado, M., LES ensembles partiellement ordonnés et le théoréme de raffinement de Schreier II, Czechoslov. Math. J., 5, 3, 308-343, (1955) · Zbl 0068.25902
[5] Cabrera, I. P.; Cordero, P.; Gutiérrez, G.; Martínez, J.; Ojeda-Aciego, M., On residuation in multilattices: filters, congruences, and homomorphisms, Fuzzy Sets Syst., 234, 1-21, (2014) · Zbl 1315.06008
[6] Cordero, P.; Gutiérrez, G.; Martínez, J.; de Guzmán, I. P., A new algebraic tool for automatic theorem provers, Ann. Math. Artif. Intell., 42, 4, 369-398, (2004) · Zbl 1095.68110
[7] Cornejo, M.; Medina, J.; Ramírez, E., A comparative study of adjoint triples, Fuzzy Sets Syst., 211, 1-14, (2013) · Zbl 1272.03111
[8] Davey, B.; Priestley, H., Introduction to lattices and order, (2002), Cambridge University Press · Zbl 1002.06001
[9] Erdos̈, P. L.; Sziklai, P.; Torney, D. C., A finite word poset, Electron. J. Comb., 8, 2, (2001), 10 pp. (electronic) · Zbl 0994.06002
[10] Ganter, B.; Kuznetsov, S. O., Stepwise construction of the Dedekind-macneille completion, (Lecture Notes in Computer Science, vol. 1453, (1998)), 295-302 · Zbl 0928.06004
[11] Ganter, B.; Wille, R., Formal concept analysis: mathematical foundation, (1999), Springer Verlag
[12] Georgescu, G.; Popescu, A., Concept lattices and similarity in non-commutative fuzzy logic, Fundam. Inform., 53, 1, 23-54, (2002) · Zbl 1023.03016
[13] Kato, R.; Watanabe, O., Substring search and repeat search using factor oracles, Inf. Process. Lett., 93, 6, 269-274, (2005) · Zbl 1173.68466
[14] Krajči, S., A generalized concept lattice, Log. J. IGPL, 13, 5, 543-550, (2005) · Zbl 1088.06005
[15] MacNeille, H., Partially ordered sets, Trans. Am. Math. Soc., 42, 416-460, (1937) · JFM 63.0833.04
[16] Medina, J.; Ojeda-Aciego, M.; Ruiz-Calviño, J., Fuzzy logic programming via multilattices, Fuzzy Sets Syst., 158, 6, 674-688, (2007) · Zbl 1111.68016
[17] Medina, J.; Ojeda-Aciego, M.; Ruiz-Calviño, J., Relating generalized concept lattices with concept lattices for non-commutative conjunctors, Appl. Math. Lett., 21, 12, 1296-1300, (2008) · Zbl 1187.06003
[18] Medina, J.; Ojeda-Aciego, M.; Ruiz-Calviño, J., Formal concept analysis via multi-adjoint concept lattices, Fuzzy Sets Syst., 160, 2, 130-144, (2009) · Zbl 1187.68589
[19] Medina, J.; Ojeda-Aciego, M.; Ruiz-Calviño, J., Concept-forming operators on multilattices, (Lecture Notes in Artificial Intelligence, vol. 7880, (2013)), 203-215 · Zbl 1396.06005
[20] Middendorf, M.; Manlove, D. F., Combined super-/substring and super-/subsequence problems, Theor. Comput. Sci., 320, 2-3, 247-267, (2004) · Zbl 1068.68114
[21] Pollandt, S., Fuzzy begriffe, (1997), Springer Berlin · Zbl 0870.06008
[22] Roman, S., Lattices and ordered sets, (2009), Springer
[23] Rudeanu, S.; Vaida, D., Revisiting the works of mihail benado, J. Mult.-Valued Log. Soft Comput., 20, 3-4, 265-307, (2013) · Zbl 1393.06001
[24] Shmuely, Z., The structure of Galois connections, Pac. J. Math., 54, 2, 209-225, (1974) · Zbl 0275.06003
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