Minimal solutions of generalized fuzzy relational equations: clarifications and corrections towards a more flexible setting. (English) Zbl 1422.03119
Summary: Computing the minimal solutions of solvable fuzzy relation equations is a very important and complex task. This computation is more complex when the algebraic structure on which the equations are defined is more general. This paper complements and corrects several results given in a recent paper, providing, as a consequence, a more general framework, in which the minimal solutions are completely determined.
MSC:
| 03E72 | Theory of fuzzy sets, etc. |
References:
| [1] | Bartl, E., Minimal solutions of generalized fuzzy relational equations: probabilistic algorithm based on greedy approach, Fuzzy Sets Syst., 260, 25-42 (2015) · Zbl 1335.03047 |
| [2] | Bartl, E.; Belohlavek, R., Hardness of solving relational equations, IEEE Trans. Fuzzy Syst., 23, 6, 2435-2438 (Dec. 2015) |
| [3] | Bělohlávek, R., Sup-t-norm and inf-residuum are one type of relational product: unifying framework and consequences, Fuzzy Sets Syst., 197, 45-58 (2012) · Zbl 1266.03056 |
| [4] | Birkhoff, G., Lattice Theory (1967), American Mathematical Society: American Mathematical Society Providence, Rhode Island · Zbl 0126.03801 |
| [5] | Chang, C.-W.; Shieh, B.-S., Linear optimization problem constrained by fuzzy max-min relation equations, Inf. Sci., 234, 71-79 (2013) · Zbl 1284.90103 |
| [6] | Ciaramella, A.; Tagliaferri, R.; Pedrycz, W.; Nola, A. D., Fuzzy relational neural network, Int. J. Approx. Reason., 41, 2, 146-163 (2006) · Zbl 1093.68078 |
| [7] | Cornejo, M. E.; Medina, J.; Ramírez-Poussa, E., Multi-adjoint algebras versus non-commutative residuated structures, Int. J. Approx. Reason., 66, 119-138 (2015) · Zbl 1350.06003 |
| [8] | Davey, B.; Priestley, H., Introduction to Lattices and Order (2002), Cambridge University Press · Zbl 1002.06001 |
| [9] | di Nola, A.; Sessa, S.; Pedrycz, W., A study on approximate reasoning mechanisms via fuzzy relation equations, Int. J. Approx. Reason., 6, 1, 33-44 (1992) · Zbl 0751.68072 |
| [10] | Díaz-Moreno, J. C.; Medina, J., Multi-adjoint relation equations: definition, properties and solutions using concept lattices, Inf. Sci., 253, 100-109 (2013) · Zbl 1320.68173 |
| [11] | Díaz-Moreno, J. C.; Medina, J., Solving systems of fuzzy relation equations by fuzzy property-oriented concepts, Inf. Sci., 222, 405-412 (2013) · Zbl 1293.68258 |
| [12] | Díaz-Moreno, J. C.; Medina, J., Using concept lattice theory to obtain the set of solutions of multi-adjoint relation equations, Inf. Sci., 266, 218-225 (2014) · Zbl 1339.03043 |
| [13] | 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) · Zbl 1393.03027 |
| [14] | Gierz, G.; Hofmann, K.; Keimel, K.; Lawson, J.; Mislove, M.; Scott, D., Continuous Lattices and Domains (2003), Cambridge University Press · Zbl 1088.06001 |
| [15] | Ignjatović, J.; Ćirić, M.; Šešelja, B.; Tepavčević, A., Fuzzy relational inequalities and equations, fuzzy quasi-orders, closures and openings of fuzzy sets, Fuzzy Sets Syst., 260, 1-24 (2015) · Zbl 1335.03053 |
| [16] | Li, D.-C.; Geng, S.-L., Optimal solution of multi-objective linear programming with inf-→ fuzzy relation equations constraint, Inf. Sci., 271, 159-178 (2014) · Zbl 1341.90150 |
| [17] | Lin, J.-L.; Wu, Y.-K.; Guu, S.-M., On fuzzy relational equations and the covering problem, Inf. Sci., 181, 14, 2951-2963 (2011) · Zbl 1231.03047 |
| [18] | Markovskii, A., On the relation between equations with max-product composition and the covering problem, Fuzzy Sets Syst., 153, 2, 261-273 (2005) · Zbl 1073.03538 |
| [19] | Perfilieva, I.; Nosková, L., System of fuzzy relation equations with inf-→ composition: complete set of solutions, Fuzzy Sets Syst., 159, 17, 2256-2271 (2008) · Zbl 1183.03053 |
| [20] | Shieh, B.-S., Solution to the covering problem, Inf. Sci., 222, 626-633 (2013) · Zbl 1293.90063 |
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