Fuzzy matroid structures (Q1175745)

Definitions fixing basic properties of a crisp matroid are given in Section 2. Its fuzzy counterpart's definition is also given together with the meaning of fuzzy rank function. Six definitions and a theorem are fixed. They are related with the results derived by the same authors in the same journal [ibid. 27, No. 3, 291-302 (1988; Zbl 0651.05024); ibid. 31, No. 3, 253-261 (1989; Zbl 0686.05013)]. Section 3 reviews properties of the theory of the sum of crisp matroids. Using the relations with the generation of sets of fuzzy matroids \((fm)\) the \(fm\)-sums are defined and the corresponding properties (generating sets of the sums, reduced forms, closeness and regularness) are derived. Five definitions and theorems are used for fixing these results. A similar task is tackled in Sections 4 and 5. The former presents results in duality and the other one in \(fm\)- product. Four definitions, five lemmas and five theorems establish the needed results.