Transitive Solutions of Relational Equations on Finite Sets and Linear Lattices

The set of solutions of relational equations over a finite referential space and with values from a linear lattice is considered. We determine in this set the greatest max-min transitive solution and the related minimal ones. Further, we investigate for the determination of particular max-min transitive solutions, namely those having Schein rank equal to 1. Related properties of convergence of fuzzy systems represented by the involved relations are also given.

[1]  Charles R. Johnson Matrix theory and applications , 1990 .

[2]  Waldemar Kolodziejczyk,et al.  DECOMPOSITION PROBLEM OF FUZZY RELATIONS—FURTHER RESULTS , 1988 .

[3]  S. Ovchinnikov STRUCTURE OF FUZZY BINARY RELATIONS , 1981 .

[4]  S. Sessa Some results in the setting of fuzzy relation equations theory , 1984 .

[5]  Sergiu Rudeanu Boolean functions and equations , 1974 .

[6]  M. Thomason Convergence of powers of a fuzzy matrix , 1977 .

[7]  Lotfi A. Zadeh,et al.  Similarity relations and fuzzy orderings , 1971, Inf. Sci..

[8]  H. Hashimoto,et al.  Convergence of powers of a fuzzy transitive matrix , 1983 .

[9]  R. Luce A note on Boolean matrix theory , 1952 .

[10]  T. S. Blyth Matrices Over Ordered Algebraic Structures , 1964 .

[11]  Elie Sanchez,et al.  Resolution of Composite Fuzzy Relation Equations , 1976, Inf. Control..

[12]  W. Pedrycz,et al.  Fuzzy Relation Equations and Their Applications to Knowledge Engineering , 1989, Theory and Decision Library.

[13]  Madan M. Gupta,et al.  Fuzzy automata and decision processes , 1977 .

[14]  Valiollah Tahani,et al.  A fuzzy model of document retrieval systems , 1976, Inf. Process. Manag..

[15]  E. Sanchez Eigen fuzzy sets and fuzzy relations , 1981 .

[16]  Ada Lettieri,et al.  Relation equations in residuated lattices , 1989 .

[17]  Witold Pedrycz,et al.  When is a fuzzy relation decomposable in two fuzzy sets , 1985 .

[18]  A. Kaufman,et al.  Introduction to the Theory of Fuzzy Subsets. , 1977 .

[19]  A. Nola Relational equations in totally ordered lattices and their complete resolution , 1985 .

[20]  H. Carter Fuzzy Sets and Systems — Theory and Applications , 1982 .

[21]  Fred W. Roush,et al.  Generalized fuzzy matrices , 1980 .

[22]  Salvatore Sessa,et al.  On the Schein rank of matrices over linear lattices , 1989 .

[23]  Ki Hang Kim Boolean matrix theory and applications , 1982 .