Fuzzy rank functions (Q1177683)

Taking \(M=(E,I)\) as a crisp matroid two definitions fix the independence of the members of \(I\) and the rank function \(R(A)\) for \(A\in I\). Two theorems, well-known results from matroid theory, are established. Similarly results for fuzzy matroids (\(fm\)) are given in Section 2. Five properties permit to obtain the fuzzy counterparts for the hypothesis used in the derivations of the two previous theorems for crisp matroids. They support the proof that a fuzzy matroid (\(fm\)) can be generated by a fuzzy rank function. Then it is shown that the \(fm\) is particulary simple. The main result establishes conditions of \(E\) and the fuzzy rank function which ensure that a fundamental sequence exists for the \(fm\). A converse theorem is obtained. These results follow from a previous paper of the same authors published in the same journal [ibid. 27, 291-302 (1988; Zbl 0651.05024)] and are connected with another one devoted to a similar derivation for \(fm\) [ibid. 41, No. 3, 343-357 (1991; see the review above)].