# zbMATH — the first resource for mathematics

Fuzzy relational equations on complete Brouwerian lattices. (English) Zbl 1248.03073
Summary: This paper deals with the problem of solving fuzzy relational equations with inf-$$\alpha _{\mathcal T}$$ composition on complete Brouwerian lattices under finite domains (resp. countably infinite domains). When the right-hand sides of fuzzy relational equations are meet-irreducible elements or have irredundant finite meet-decompositions, some necessary and sufficient conditions for attainable solutions (resp. unattainable solutions) are formulated and some properties of the attainable solutions (resp. the unattainable solutions) are shown. Further, the solution sets are given when they are nonempty.

##### MSC:
 03E72 Theory of fuzzy sets, etc. 06B23 Complete lattices, completions 06D20 Heyting algebras (lattice-theoretic aspects)
Full Text:
##### References:
 [1] De Baets, B., Analytical solution methods for fuzzy relational equations, (), 291-340 · Zbl 0970.03044 [2] Birkhoff, G., Lattice theory, (1979), American Mathematical Society Colloquium Publications Providence, R.I., vol, XXV · Zbl 0126.03801 [3] Crawley, P.; Dilworth, R.P., Algebraic theory of lattices, (1973), Prentice-Hall Englewood Cliffs, NJ · Zbl 0494.06001 [4] Di Nola, A.; Sessa, S.; Pedrycz, W.; Higashi, M., Minimal and maximal solutions of a decomposition problem of fuzzy relation, International journal of general systems, 11, 103-116, (1985) [5] Di Nola, A.; Sessa, S.; Pedrycz, W.; Sanchez, E., Fuzzy relation equations and their applications to knowledge engineering, (1989), Kluwer Academic Publishers Dordrecht · Zbl 0694.94025 [6] Imai, H.; Kikuchi, K.; Miyakoshi, M., Unattainable solutions of a fuzzy relation equation, Fuzzy sets and systems, 99, 193-196, (1998) · Zbl 0938.03081 [7] Li, Y.M.; Wang, X.P., Necessary and sufficient conditions for existence of maximal solutions for inf-α composite fuzzy relational equations, Computers and mathematics with applications, 55, 1961-1973, (2008) · Zbl 1138.03322 [8] Li, Y.M.; Wang, X.P., The solution sets of -fuzzy relational equations in finite domains and on a complete Brouwerian lattice, Indian journal of pure & applied mathematics, 34, 1249-1257, (2003) · Zbl 1053.03513 [9] Loia, V.; Sessa, S., Fuzzy relation equations for coding/decoding processes of images and videos, Information sciences, 171, 145-172, (2005) · Zbl 1078.68815 [10] Mizumoto, M.; Zimmermann, H.J., Comparison of fuzzy reasoning methods, Fuzzy sets and systems, 8, 253-283, (1982) · Zbl 0501.03013 [11] Nobuhara, H.; Pedrycz, W.; Sessa, S.; Hirota, K., A motion compression/reconstruction method based on MAX t-norm composite fuzzy relational equations, Information sciences, 176, 2526-2552, (2006) · Zbl 1102.68698 [12] Perfilieva, I.; Nosková, L., System of fuzzy relation equations with inf-→ composition: complete set of solutions, Fuzzy sets and systems, 159, 2256-2271, (2008) · Zbl 1183.03053 [13] Qu, X.B.; Wang, X.P., Some properties of infinite fuzzy relational equations on complete Brouwerian lattices, Fuzzy sets and systems, 158, 1327-1339, (2007) · Zbl 1120.03041 [14] Sanchez, E., Resolution of composite fuzzy relation equations, Information and control, 30, 38-48, (1976) · Zbl 0326.02048 [15] Shieh, B.S., Infinite fuzzy relation equations with continuous t-norms, Information sciences, 178, 1961-1967, (2008) · Zbl 1135.03346 [16] Wang, X.P.; Xia, C., The solution sets of infinite fuzzy relational equations with sup-conjunctor composition complete distributive lattices, Fuzzy sets and systems, 160, 2989-3006, (2009) · Zbl 1183.03056 [17] Wang, X.P., Method of solution to fuzzy relation equations in a complete Brouwerian lattice, Fuzzy sets and systems, 120, 409-414, (2001) · Zbl 0981.03055 [18] Wang, X.P.; Qu, X.B., Continuous join-irreducible elements and their applications to describe the solution set of fuzzy relational equations, Acta Mathematica sinica (Chinese series), 49, 1171-1180, (2006), (in Chinese) · Zbl 1120.03042 [19] Wang, Z.D.; Yu, Y.D., Pseudo-t-norms and implication operations on a complete Brouwerian lattice, Fuzzy sets and systems, 132, 113-124, (2002) · Zbl 1013.03020 [20] Wang, Z.D.; Dai, F.M., The sets of solutions of L-relation equations T(a,x)=b and I(a,x)=b, Journal of yangzhou university, 6, 8-10, (2003), (in Chinese) [21] Wu, W.M., The inf-α compositioin of fuzzy relations, Journal of Shanghai normal university (natural), 2, 1-7, (1985), (in Chinese) · Zbl 0633.54005 [22] Xiong, Q.Q.; Wang, X.P., Some properties of sup-MIN fuzzy relational equations on infinite domains, Fuzzy sets and systems, 151, 393-402, (2005) · Zbl 1062.03053 [23] Xiong, Q.Q.; Wang, X.P., Solution sets of $$\inf - \alpha_{\mathcal{T}}$$ fuzzy relational equations on complete Brouwerian lattices, Information sciences, 177, 4757-4767, (2007) · Zbl 1129.03032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.