Fuzzy matrices of permanent one. (English) Zbl 0796.15019
The paper deals with square matrices over the semiring \(([0,1], \max, \min)\). Matrices with permanent one [max-min determinant, cf. J. B. Kim, A. Baartmans and N. S. Sahadin, ibid. 29, No. 3, 349- 356 (1989; Zbl 0668.15004)] are considered as a generalization of reflexive fuzzy relations on a finite set [cf. A. Kaufmann, Fuzzy sets theory (1975; Zbl 0456.68081)]. The notion of the generalized inverse [cf. P. S. S. N. V. Prasada Rao, K. P. S. Bhaskara Rao, Linear Algebra Appl. 11, 135-153 (1975; Zbl 0322.15011)] is characterized for these matrices and applied to study of idempotent matrices and fuzzy linear mappings.
Reviewer: J.Drewniak (Katowice)
MSC:
| 15B33 | Matrices over special rings (quaternions, finite fields, etc.) |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 15A09 | Theory of matrix inversion and generalized inverses |
| 94D05 | Fuzzy sets and logic (in connection with information, communication, or circuits theory) |
| 03E72 | Theory of fuzzy sets, etc. |
Keywords:
max-min determinant; permanent; reflexive fuzzy relations; generalized inverse; idempotent matrices; fuzzy linear mappingsReferences:
| [1] | Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences (1979), Academic: Academic New York · Zbl 0484.15016 |
| [2] | Cao, Z.-Q.; Kim, K. H.; Rouch, F. W., Incline Algebra and Applications (1984), Eillis Horwood: Eillis Horwood Chichester · Zbl 0541.06009 |
| [3] | H.H. Cho, Regular matrices in the semigroup of Hall matrices, To appear in Linear Algebra Appl.; H.H. Cho, Regular matrices in the semigroup of Hall matrices, To appear in Linear Algebra Appl. · Zbl 0787.05018 |
| [4] | Kandel, A., Fuzzy Mathematical Techniques with Applications (1986), Addison-Wesley: Addison-Wesley Reading · Zbl 0668.94022 |
| [5] | Kim, J. B.; Baartmans, A.; Sahadin, N. S., Determinant theory for fuzzy matrices, Fuzzy Sets and Systems, 29, 349-356 (1989) · Zbl 0668.15004 |
| [6] | Kim, K. H.; Roush, F. W., Generalized fuzzy matrices, Fuzzy Sets and Systems, 4, 293-315 (1980) · Zbl 0451.20055 |
| [7] | Prasada Rao, P. S.S. N.V.; Bhaskara Rao, K. P.S., On generalized inverses of Boolean matrices, Linear Algebra and Appl., 11, 135-153 (1975) · Zbl 0322.15011 |
| [8] | Schein, B. M., Regular elements of the semigroup of all binary relations, Semigroup Forum, 13, 95-102 (1976) · Zbl 0355.20058 |
| [9] | E. Sanchez, Eigen fuzzy sets and fuzzy relations, J. Math. Anal Appl.81; E. Sanchez, Eigen fuzzy sets and fuzzy relations, J. Math. Anal Appl.81 · Zbl 0466.04003 |
| [10] | Thornton, M. C.; Hardy, D. W., The intersection of the maximal regular subsemigroups of the semigroup of binary relations, Semigroup Forum, 29, 343-349 (1984) · Zbl 0536.20042 |
| [11] | Zhao, C.-K., Inverses of L-fuzzy matrices, Fuzzy Sets and Systems, 34, 103-116 (1990) · Zbl 0687.15011 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
