zbMATH — the first resource for mathematics

Unique solvability of max-min fuzzy equations and strong regularity of matrices over fuzzy algebra. (English) Zbl 0852.15011
Summary: A matrix is said to have strongly linearly independent columns (or, in the case of square matrices, to be strongly regular) if for some vector \(b\) the system \(A\otimes x=b\) has a unique solution. We formulate a necessary and sufficient condition for a system of linear equations over a fuzzy algebra to have a unique solution and prove the equivalence of strong regularity and the trapezoidal property. Moreover, an algorithm for testing these properties is discussed.

15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A06 Linear equations (linear algebraic aspects)
03E72 Theory of fuzzy sets, etc.
Full Text: DOI
[1] Butkovič, P., Strong regularity of matrices - a survey of results, Discrete appl. math., 48, 45-68, (1994) · Zbl 0804.06017
[2] Butkovič, P.; Cechlárová, K.; Szabó, P., Strong linear independence in bottleneck algebra, Linear algebra appl., 94, 133-155, (1987) · Zbl 0629.90093
[3] Butkovič, P.; Hevery, F., A condition for the strong regularity of matrices in the minimax algebra, Discrete appl. math., 11, 209-222, (1985) · Zbl 0602.90136
[4] Butkovič, P.; Szabó, P., An algorithm for checking strong regularity of matrices in the bottleneck algebra, ()
[5] Cechlárová, K., Strong regularity of matrices in a discrete bottleneck algebra, Linear algebra appl., 128, 35-50, (1990) · Zbl 0704.15003
[6] Cuninghame-Green, R.A., Minimax algebra, () · Zbl 0498.90084
[7] Guo, S.Z.; Wang, P.Z.; Di Nola, A.; Sesa, S., Further contributions to the study of finite fuzzy relation equations, Fuzzy sets and systems, 26, 93-104, (1988) · Zbl 0645.04003
[8] Higashi, M.; Klir, G.J., Resolution of finite fuzzy relation equations, Fuzzy sets and systems, 13, 65-82, (1984) · Zbl 0553.04006
[9] Jian-Xin, Li, The smallest solution of MAX-MIN fuzzy equations, Fuzzy sets and systems, 41, 317-327, (1990) · Zbl 0731.04006
[10] Sanchez, E., Resolution of composite fuzzy relation equations, Inform. and control, 30, 38-48, (1976) · Zbl 0326.02048
[11] Vorobyev, N.N., Extremal algebra of positive matrices, Elektronische informationsverarbeitung und kybernetik, 3, (1967), (in Russian)
[12] Zimmermann, K., Extremal algebra, (1976), Ekon. ústav ČSAV Praha, (in Czech) · Zbl 0438.90102
[13] Zimmermann, U., Linear and combinatorial optimization in ordered algebraic structures, () · Zbl 0466.90045
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.