# zbMATH — the first resource for mathematics

Some properties of infinite fuzzy relational equations on complete Brouwerian lattices. (English) Zbl 1120.03041
Summary: This paper deals with fuzzy relational equations $$A\odot X=b$$ (where both $$A=(a_j)_{j\in J}$$ and $$b$$ are known, $$b$$ has an irredundant continuous join-decomposition, $$X=(x_j)^T_{j\in J}$$ is unknown, “$$\odot$$” represents “sup-inf” composition, $$J$$ is an infinite index set) on complete Brouwerian lattices. At first, a necessary and sufficient condition for the existence of the attainable solutions (the unattainable solutions, respectively) and some properties of the attainable solutions (the unattainable solutions, respectively) are formulated. Then the solution set of the fuzzy relational equations is described.

##### MSC:
 3e+72 Theory of fuzzy sets, etc.
Full Text:
##### References:
  G. Birkhoff, Lattice Theory, vol. XXV, third ed., American Mathematical Society Colloquium Publications, Providence, RI, 1979.  Chen, L.; Wang, P.P., Fuzzy relation equations (I): the general and specialized solving algorithms, Soft computing, 6, 428-435, (2002) · Zbl 1024.03520  Crawley, P.; Dilworth, R.P., Algebraic theory of lattices, (1973), Prentice-Hall Englewood Cliffs, NJ · Zbl 0494.06001  De Baets, B., An order-theoretic approach to solving sup-T equations, (), 67-87 · Zbl 0874.04005  De Baets, B., Analytical solution methods for fuzzy relational equations, (), 291-340 · Zbl 0970.03044  Di Nola, A., On solving relational equations in Brouwerian lattices, Fuzzy sets and systems, 34, 365-376, (1990) · Zbl 0701.04003  Di Nola, A.; Sessa, S.; Pedrycz, W.; Higashi, M., Minimal and maximal solutions of a decomposition problem of fuzzy relations, Internat. J. general systems, 11, 103-116, (1985)  Di Nola, A.; Sessa, S.; Pedrycz, W.; Sanchez, E., Fuzzy relation equations and their applications to knowledge engineering, (1989), Kluwer Academic Publishers Dordrecht, Boston, London · Zbl 0694.94025  Higashi, M.; Klir, G.J., Resolution of finite fuzzy relation equations, Fuzzy sets and systems, 13, 65-82, (1984) · Zbl 0553.04006  Imai, H.; Kikuchi, K.; Miyakoshi, M., Unattainable solutions of a fuzzy relation equation, Fuzzy sets and systems, 99, 193-196, (1998) · Zbl 0938.03081  Imai, H.; Miyakoshi, M.; Da-te, T., Some properties of minimal solutions for a fuzzy relation equation, Fuzzy sets and systems, 90, 335-340, (1997) · Zbl 0919.04008  Kim, K.H., Boolean matrix theory and applications, (1982), Marcel Dekker New York  Li, J.X., On an algorithm for solving fuzzy linear systems, Fuzzy sets and systems, 61, 369-371, (1994) · Zbl 0826.04004  Luo, C.Z., Reachable solution set of a fuzzy relation equation, J. math. anal. appl., 103, 524-532, (1984) · Zbl 0588.04005  Peeva, K., Fuzzy linear systems, Fuzzy sets and systems, 49, 339-355, (1992) · Zbl 0805.04005  Peeva, K.; Kyosev, Y., Fuzzy relational calculus-theory, applications and software, (2004), World Scientific Publishing Company Singapore · Zbl 1083.03048  Sanchez, E., Resolution of composite fuzzy relation equations, Inform. and control, 30, 38-48, (1976) · Zbl 0326.02048  Wang, X.P., A method to solve a fuzzy relational equation in a lattice $$[0, 1]$$, Appl. math. J. Chinese univ. ser. A, 15, 127-133, (2000), (in Chinese) · Zbl 0962.03049  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  Wang, X.P., Conditions under which a fuzzy relational equation has minimal solution in a complete Brouwerian lattice, Adv. in math., 31, 220-228, (2002), (in Chinese) · Zbl 1264.03113  Wang, X.P., Infinite fuzzy relational equations on a complete Brouwerian lattice, Fuzzy sets and systems, 138, 657-666, (2003) · Zbl 1075.03026  Wang, X.P.; Qu, X.B., Continuous join-irreducible elements and their applications to describe the solution set of fuzzy relational equations, Acta math. sinica, Chinese series, 49, 1171-1180, (2006), (in Chinese) · Zbl 1120.03042  Zhao, C.K., On matrix equations in a class of complete and completely distributive lattice, Fuzzy sets and systems, 22, 303-320, (1987) · Zbl 0621.06006
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.