Abstract
In this paper, we focus on a twofold relational generalization of the notion of Galois connection. It is twofold because it is defined between sets endowed with arbitrary transitive relations and, moreover, both components of the connection are relations as well. Specifically, we introduce the notion of relational Galois connection between two transitive digraphs, study some of its properties and its relationship with other existing approaches in the literature.
Partially supported by the Spanish research projects TIN15-70266-C2-P-1, PGC2018-095869-B-I00 and TIN2017-89023-P of the Science and Innovation Ministry of Spain and the European Social Fund.
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Notes
- 1.
Notice that, as usual, we use the same symbol to denote both binary relations which need not be equal.
- 2.
A digraph is often called a relational system.
References
Abramsky, S.: Big toy models: representing physical systems as Chu spaces. Synthese 186(3), 697–718 (2012)
Antoni, L., Krajči, S., Krídlo, O.: Representation of fuzzy subsets by Galois connections. Fuzzy Sets Syst. 326, 52–68 (2017)
Cabrera, I.P., Cordero, P., García-Pardo, F., Ojeda-Aciego, M., De Baets, B.: On the construction of adjunctions between a fuzzy preposet and an unstructured set. Fuzzy Sets Syst. 320, 81–92 (2017)
Cabrera, I.P., Cordero, P., García-Pardo, F., Ojeda-Aciego, M., De Baets, B.: Adjunctions between a fuzzy preposet and an unstructured set with underlying fuzzy equivalence relations. IEEE Trans. Fuzzy Syst. 26(3), 1274–1287 (2018)
Cabrera, I.P., Cordero, P., Ojeda-Aciego, M.: Relation-based Galois connections: towards the residual of a relation. In: Proceedings of Computational and Mathematical Methods in Science and Engineering (CMMSE 2017) (2017)
Denniston, J.T., Melton, A., Rodabaugh, S.E.: Formal contexts, formal concept analysis, and Galois connections. Electr. Proc. Theor. Comput. Sci. 129, 105–120 (2013)
Domenach, F., Leclerc, B.: Biclosed binary relations and Galois connections. Order 18(1), 89–104 (2001)
Ganter, B., Wille, R.: Formal Concept Analysis. Springer, Heidelberg (1999). https://doi.org/10.1007/978-3-642-59830-2
Ganter, B.: Relational Galois connections. In: Kuznetsov, S.O., Schmidt, S. (eds.) ICFCA 2007. LNCS (LNAI), vol. 4390, pp. 1–17. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-70901-5_1
García-Pardo, F., Cabrera, I.P., Cordero, P., Ojeda-Aciego, M.: On Galois connections and soft computing. In: Rojas, I., Joya, G., Cabestany, J. (eds.) IWANN 2013. LNCS, vol. 7903, pp. 224–235. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38682-4_26
García-Pardo, F., Cabrera, I.P., Cordero, P., Ojeda-Aciego, M.: On closure systems and adjunctions between fuzzy preordered sets. In: Baixeries, J., Sacarea, C., Ojeda-Aciego, M. (eds.) ICFCA 2015. LNCS (LNAI), vol. 9113, pp. 114–127. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-19545-2_7
García-Pardo, F., Cabrera, I.P., Cordero, P., Ojeda-Aciego, M., Rodríguez-Sanchez, F.J.: On the existence of isotone Galois connections between preorders. In: Glodeanu, C.V., Kaytoue, M., Sacarea, C. (eds.) ICFCA 2014. LNCS (LNAI), vol. 8478, pp. 67–79. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-07248-7_6
García, F., Cabrera, I.P., Cordero, P., Ojeda-Aciego, M., Rodriguez, F.: On the definition of suitable orderings to generate adjunctions over an unstructured codomain. Inf. Sci. 286, 173–187 (2014)
Jeřábek, E.: Galois connection for multiple-output operations. Algebra Univers. 79, 17 (2018)
Krídlo, O., Ojeda-Aciego, M.: Formal concept analysis and structures underlying quantum logics. In: Medina, J., et al. (eds.) IPMU 2018. CCIS, vol. 853, pp. 574–584. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-91473-2_49
Krídlo, O., Ojeda-Aciego, M.: Relating Hilbert-Chu correspondences and big toy models for quantum mechanics. In: Kóczy, L., Medina-Moreno, J., Ramírez-Poussa, E., Šostak, A. (eds.) Computational Intelligence and Mathematics for Tackling Complex Problems. SCI, vol. 819, pp. 75–80. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-16024-1_10
Ore, O.: Galois connexions. Trans. Am. Math. Soc. 55(3), 493–513 (1944)
Wille, R.: Subdirect product constructions of concept lattices. Discrete Math. 63, 305–313 (1987)
Xia, W.: Morphismen als formale Begriffe-Darstellung und Erzeugung. Verlag Shaker, Aachen (1993)
Zhang, G.-Q.: Logic of Domains. Springer, Boston (1991). https://doi.org/10.1007/978-1-4612-0445-9
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Cabrera, I.P., Cordero, P., Muñoz-Velasco, E., Ojeda-Aciego, M. (2019). A Relational Extension of Galois Connections. In: Cristea, D., Le Ber, F., Sertkaya, B. (eds) Formal Concept Analysis. ICFCA 2019. Lecture Notes in Computer Science(), vol 11511. Springer, Cham. https://doi.org/10.1007/978-3-030-21462-3_19
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