This paper attempts to study circuit axioms and connectedness of M-fuzzifying matroids. Based on the results of circuit-maps given by Yao and Shi, remarks concerning circuit-maps and connectedness of fuzzifying matroids are shown, and thus the useful notion of an M-fuzzifying matroid induced by an M-fuzzy family of circuits is introduced. Circuit axioms of an M-fuzzifying matroid induced by an M-fuzzy family of circuits are established, which are the generalizations of circuit axioms for a matroid in a normal way. Connectedness of an M-fuzzifying matroid induced by an M-fuzzy family of circuits is then defined by its M-fuzzy family of circuits, a fuzzy analog of connectedness of a matroid. Additionally, two kinds of M-fuzzifying matroids are compared.
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