Abstract
Algebraic structures are essential in fuzzy frameworks such as fuzzy formal concept analysis and fuzzy rough set theory. This paper studies two general structures such as right adjoint algebras and operator left residuated posets, introducing several properties which relate them. Different extensions of the operators included in a given operator left residuated poset are presented and a reasoned analysis is shown to guarantee that the equivalence satisfied by the operators in this structure is not a generalization of the usual adjoint property, which is a basic property verified by right adjoint pairs. Operator left residuated posets are also studied in the framework of the Dedekind-MacNeille completion of a poset.
This work is partially supported by the 2014-2020 ERDF Operational Programme in collaboration with the State Research Agency (AEI) in projects TIN2016-76653-P and PID2019-108991GB-I00, and with the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia in project FEDER-UCA18-108612, and by the European Cooperation in Science & Technology (COST) Action CA17124.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Alfonso-Robaina, D., Díaz-Moreno, J.C., Malleuve-Martinez, A., Medina, J., Rubio-Manzano, C.: Modeling enterprise architecture and strategic management from fuzzy decision rules. Stud. Comput. Intell. 819, 139–147 (2020)
Antoni, L., Cornejo, M.E., Medina, J., Ramirez-Poussa, E.: Attribute classification and reduct computation in multi-adjoint concept lattices. IEEE Trans. Fuzzy Syst. 29(5), 1121–1132 (2020)
Antoni, L., Krajči, S., Krídlo, O.: Constraint heterogeneous concept lattices and concept lattices with heterogeneous hedges. Fuzzy Sets Syst. 303, 21–37 (2016)
Antoni, L., Krajči, S., Krídlo, O.: On stability of fuzzy formal concepts over randomized one-sided formal context. Fuzzy Sets Syst. 333, 36–53 (2018)
Bělohlávek, R.: Lattices of fixed points of fuzzy Galois connections. Math. Log. Q. 47(1), 111–116 (2001)
Bělohlávek, R.: Concept lattices and order in fuzzy logic. Ann. Pure Appl. Log. 128, 277–298 (2004)
Benítez-Caballero, M.J., Medina, J., Ramírez-Poussa, E., Ślȩzak, D.: Rough-set-driven approach for attribute reduction in fuzzy formal concept analysis. Fuzzy Sets Syst. 319, 117–138 (2020)
Benítez-Caballero, M.J., Medina, J., Ramírez-Poussa, E., Ślȩzak, D.: Bireducts with tolerance relations. Inf. Sci. 435, 26–39 (2018)
Chajda, I.: A representation of residuated lattices satisfying the double negation law. Soft. Comput. 22(6), 1773–1776 (2017). https://doi.org/10.1007/s00500-017-2673-9
Chajda, I., Länger, H.: Left residuated operators induced by posets with a unary operation. Soft. Comput. 23(22), 11351–11356 (2019). https://doi.org/10.1007/s00500-019-04028-w
Chajda, I., Länger, H.: Residuation in modular lattices and posets. Asian-Eur. J. Math. 12(02), 1950092 (2019)
Cornejo, M.E., Díaz-Moreno, J.C., Medina, J.: Multi-adjoint relation equations: a decision support system for fuzzy logic. Int. J. Intell. Syst. 32(8), 778–800 (2017)
Cornejo, M.E., Lobo, D., Medina, J.: Solving generalized equations with bounded variables and multiple residuated operators. Math. 8(11), 1992 (2020). 1–22
Cornejo, M.E., Lobo, D., Medina, J.: Extended multi-adjoint logic programming. Fuzzy Sets Syst. 388, 124–145 (2020)
Cornejo, M.E., Medina, J., Ramírez-Poussa, E.: A comparative study of adjoint triples. Fuzzy Sets Syst. 211, 1–14 (2013)
Cornejo, M.E., Medina, J., Ramírez-Poussa, E.: Multi-adjoint algebras versus extended-order algebras. Appl. Math. Inf. Sci. 9(2L), 365–372 (2015)
Cornejo, M.E., Medina, J., Ramírez-Poussa, E.: Multi-adjoint algebras versus non-commutative residuated structures. Int. J. Approx. Reason. 66, 119–138 (2015)
Cornejo, M.E., Medina, J., Ramírez-Poussa, E.: Algebraic structure and characterization of adjoint triples, Fuzzy Sets and Systems, 9 February 2021 (In Press)
Cornejo, M.E., Medina, J., Ramírez-Poussa, E.: Implication operators generating pairs of weak negations and their algebraic structure. Fuzzy Sets Syst. 405, 18–39 (2021)
Cornelis, C., Medina, J., Verbiest, N.: Multi-adjoint fuzzy rough sets: definition, properties and attribute selection. Int. J. Approx. Reason. 55, 412–426 (2014)
Damásio, C., Medina, J., Ojeda-Aciego, M.: Termination of logic programs with imperfect information: applications and query procedure. J. Appl. Log. 5, 435–458 (2007)
Damásio, C.V., Pereira, L.M.: Monotonic and residuated logic programs. In: Benferhat, S., Besnard, P. (eds.) ECSQARU 2001. LNCS (LNAI), vol. 2143, pp. 748–759. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44652-4_66
Davey, B., Priestley, H.: Introduction to Lattices and Order, 2nd Edition, Cambridge University Press, Cambridge (2002)
Díaz-Moreno, J.C., Medina, J.: Multi-adjoint relation equations: definition, properties and solutions using concept lattices. Inf. Sci. 253, 100–109 (2013)
Díaz-Moreno, J.C., Medina, J., Turunen, E.: Minimal solutions of general fuzzy relation equations on linear carriers. an algebraic characterization. Fuzzy Sets Syst. 311, 112–123 (2017)
Dilworth, R.P., Ward, M.: Residuated lattices. Trans. Am. Math. Soc. 45, 335–354 (1939)
Hájek, P.: Metamathematics of Fuzzy Logic, Trends in Logic. Kluwer Academic, Dordrecht (1998)
Julián-Iranzo, P., Medina, J., Ojeda-Aciego, M.: On reductants in the framework of multi-adjoint logic programming. Fuzzy Sets Syst. 317, 27–43 (2017)
Julián, P., Moreno, G., Penabad, J.: On fuzzy unfolding: a multi-adjoint approach. Fuzzy Sets Syst. 154(1), 16–33 (2005)
Lobo, D., Cornejo, M.E., Medina, J.: Abductive reasoning in normal residuated logic programming via bipolar max-product fuzzy relation equations. In: Conference of the International Fuzzy Systems Association and the European Society for Fuzzy Logic and Technology (EUSFLAT 2019), vol. 1 of Atlantis Studies in Uncertainty Modelling, pp. 588–594. Atlantis Press (2019)
Madrid, N., Ojeda-Aciego, M.: On the measure of incoherent information in extended multi-adjoint logic programs. In: Proceedings of the 2013 IEEE Symposium on Foundations of Computational Intelligence (FOCI), Institute of Electrical and Electronics Engineers (IEEE) (2013)
Medina, J.: Multi-adjoint property-oriented and object-oriented concept lattices. Inf. Sci. 190, 95–106 (2012)
Medina, J., Ojeda-Aciego, M., Ruiz-Calviño, J.: Formal concept analysis via multi-adjoint concept lattices. Fuzzy Sets Syst. 160(2), 130–144 (2009)
Medina, J., Ojeda-Aciego, M., Vojtáš, P.: A multi-adjoint logic approach to abductive reasoning. In: Codognet, P. (ed.) ICLP 2001. LNCS, vol. 2237, pp. 269–283. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45635-X_26
Moreno, G., Penabad, J., Riaza, J.A.: Symbolic Unfolding of Multi-adjoint Logic Programs, pp. 43–51. Springer International Publishing, Cham (2019)
Moreno, G., Penabad, J., Vázquez, C.: Beyond multi-adjoint logic programming. Int. J. Comput. Math. 92(9), 1956–1975 (2015)
Rubio-Manzano, C., Díaz-Moreno, J.C., Alfonso-Robaina, D., Malleuve, A., Medina, J.: A novel cause-effect variable analysis in enterprise architecture by fuzzy logic techniques. Int. J. Comput. Intell. Syst. 13, 511–523 (2020)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Cornejo, M.E., Medina, J. (2021). Right Adjoint Algebras Versus Operator Left Residuated Posets. In: Ramanna, S., Cornelis, C., Ciucci, D. (eds) Rough Sets. IJCRS 2021. Lecture Notes in Computer Science(), vol 12872. Springer, Cham. https://doi.org/10.1007/978-3-030-87334-9_15
Download citation
DOI: https://doi.org/10.1007/978-3-030-87334-9_15
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-87333-2
Online ISBN: 978-3-030-87334-9
eBook Packages: Computer ScienceComputer Science (R0)