Rough set theory and concept lattices to solve
fuzzy relation equations?
Jesús Medina1[0000−0002−3931−5873]
Department of Mathematics. University of Cádiz, Spainjesus.medina@uca.es
https://www.mcis.uca.es

Property-oriented concept lattices and object-oriented concept lattices [5, 11,14] arose as a generalization of rough set theory (RST) considering the philosophy of formal concept analysis (FCA) [13]. These frameworks were introducedin the multi-adjoint fuzzy setting [6] in the paper [17]. Specifically, the lower andupper approximation operators are generalized to be applied to a quantitative
relationship between the objects and attributes (informative table) instead ofits discernibility relation [8, 17]. The new (modal) operators are called necessityand possibility operators, respectively.
Fuzzy relation equations (FRE) are associated with the composition of fuzzy
relations and is an appropriate tool for handling and modeling non-probabilistic
forms of uncertainty. From their introduction in the eighties by Elie Sanchez [19],FRE have received great interest from their introduction due to the wide rangeof their applications in different areas of real-world, such as decision making [7],bipolarity [12], optimization [2] and image processing [1].
A relevant achievement was to characterize the solvability of FRE with thecomputation of concepts in a property-oriented concept lattice or an objectoriented concept lattice [9, 10]. Given a FRE, depending on whether the unknown relation is to the left side or to the right side, we have determined an
associated property-oriented context or an object-oriented context, respectively,
whose concepts provide solvable FREs related to the original one [9].
This relationship has allowed diverse advances in FRE. For example, it has
been possible to introduce three different mechanisms of approximating unsolvable FRE. Two of them are based on the computation of closest concepts associated with a given fuzzy subset of attributes [7] and the third one is basedon the theory of attribute reduction [3, 4, 18] developed in the mentioned concept lattices frameworks [16, 15]. This last theory has also allow the reductionof equations of the system associated with a given FRE [15], which is a novel
procedure that had not been envisaged until now.
This contribution will be focused on these relations and achievements, andwill present new challenges.
?

Partially supported by the projects PID2019-108991GB-I00 and PID2022137620NB-I00/AEI/10.13039/501100011033/ FEDER, UE, and with grant
TED2021-129748B-I00 funded by MCIN/AEI/10.13039/501100011033 and, asappropriate, by “ERDF A way of making Europe”, by the “European Union” or by
the “European Union NextGenerationEU/PRTR”.

2Jesús Medina

References
1. C. Alcalde, A. Burusco, J. C. Díaz-Moreno, and J. Medina. Fuzzy concept lattices
and fuzzy relation equations in the retrieval processing of images and signals.
Inter. J. of Uncertainty, Fuzziness and Knowledge-Based Systems, 25:99–120, 2017.
2. S. Aliannezhadi and A. Abbasi Molai. A new algorithm for geometric optimization with a single-term exponent constrained by bipolar fuzzy relation equations.
Iranian Journal of Fuzzy Systems, 18(1):137–150, 2021.
3. L. Antoni, M. E. Cornejo, J. Medina, and E. Ramirez. Attribute classification andreduct computation in multi-adjoint concept lattices. IEEE Transactions on FuzzySystems, 29:1121–1132, 2021.
4. M. J. Benítez-Caballero, J. Medina, E. Ramírez-Poussa, and D. Ślȩzak. Roughset-driven approach for attribute reduction in fuzzy formal concept analysis. Fuzzy
Sets and Systems, 391:117–138, 2020.
5. Y. Chen and Y. Yao. A multiview approach for intelligent data analysis based ondata operators. Information Sciences, 178(1):1–20, 2008.
6. M. E. Cornejo and J. Medina. Impact Zadeh’s theory to algebraic structures. multiadjoint algebras. Journal of Pure and Applied Mathematics, 12:126–141, 2021.
7. M. E. Cornejo, J. Medina, and E. Ramírez-Poussa. Attribute and size reduction
mechanisms in multi-adjoint concept lattices. Journal of Computational and Applied Mathematics, 318:388 – 402, 2017.
8. C. Cornelis, J. Medina, and N. Verbiest. Multi-adjoint fuzzy rough sets: Definition, properties and attribute selection. International Journal of Approximate
Reasoning, 55:412–426, 2014.
9. J. C. Díaz-Moreno and J. Medina. Multi-adjoint relation equations: Definition,properties and solutions using concept lattices. Information Sciences, 253:100–109, 2013.
10. J. C. Díaz-Moreno and J. Medina. Using concept lattice theory to obtain the setof solutions of multi-adjoint relation equations. Information Sciences, 266(0):218–225, 2014.
11. I. Düntsch and G. Gediga. Approximation operators in qualitative data analysis.
In Theory and Applications of Relational Structures as Knowledge Instruments,pages 214–230, 2003.
12. S. Freson, B. De Baets, and H. De Meyer. Linear optimization with bipolar maxmin constraints. Information Sciences, 234:3 – 15, 2013. Fuzzy Relation Equations:
New Trends and Applications.
13. B. Ganter and R. Wille. Formal Concept Analysis: Mathematical Foundation.
Springer Verlag, 1999.
14. G. Gediga and I. Düntsch. Modal-style operators in qualitative data analysis. InProc. IEEE Int. Conf. on Data Mining, pages 155–162, 2002.
15. D. Lobo, V. López-Marchante, and J. Medina. Reducing fuzzy relation equations
via concept lattices. Fuzzy Sets and Systems, 463:108465, 2023.
16. D. Lobo, V. López-Marchante, and J. Medina. On measuring the solvabilityof a fuzzy relation equation. Studies in Computational Intelligence, 1127:9-15, 2024.
17. J. Medina. Multi-adjoint property-oriented and object-oriented concept lattices.
Information Sciences, 190:95–106, 2012.
18. J. Medina.
Relating attribute reduction in formal, object-oriented andproperty-oriented concept lattices. Computers & Mathematics with Applications,64(6):1992–2002, 2012.
19. E. Sanchez. Resolution of composite fuzzy relation equations. Information andControl, 30(1):38–48, 1976.