Closure axioms for a class of fuzzy matroids and co-towers of matroids

This paper represents an attempt to define a closure operator which can determine Geotschel and Voxman fuzzy matroid (GV fuzzy matroid for short), and presents an application of the notion co-tower in GV fuzzy matroids. Two bijections are obtained for a given finite set X, one is from to BX (the set of all triples of operators on X satisfying some fuzzy closure axioms), the other is from FM(X) (the set of all systems of GV independent fuzzy sets on X) to Mc(X) (the set of all co-tower structures on X). The first bijection shows that a triple of operators on X satisfying some fuzzy closure axioms can determine a closed singular GV fuzzy matroid or a closed proper single GV fuzzy matroid, and the second shows that a co-tower structure on X can determine a GV fuzzy matroid. As a result, it is proved that the category FM of all GV fuzzy matroids and continuous mappings is isomorphic to the category Mc of all co-towers in M (the category of all matroids and continuous mappings) and continuous mappings.