| The theory of convex spaces via abstracting the basic properties of convex sets is a mathematical branch which deals with set systems.Convex structures are closely related to matroid theory and topological structures,and they have many common features.With the development of fuzzy topology and fuzzy matroid theory,many kinds of fuzzy convex structures have been investigated.At present,the most representative fuzzy convex structures include L-convex structures,M-fuzzifying convex structures and(L,M)-fuzzy convex structures.The research on fuzzy convex space theory is mainly based on these three frameworks.Among this three kinds of fuzzy convex structures,(L,M)-fuzzy convex structure is the most general one in sense that classical convex structure,L-convex structure and M-fuzzifying convex structure are the special cases of(L,M)-fuzzy convex structure.Because of the double fuzziness of the(L,M)-fuzzy convex structure,its study is more complicated.In this paper,the separation axioms in M-fuzzifying convex spaces and(L,M)-fuzzy convex spaces are studied.First of all,we introduce the degree to which an M-fuzzifying convex space is S0,S1 and S2 separated.Each M-fuzzifying convex space can be regarded to be S0,Si and S2 separated to some degree.Therefore,they are more in line with the inherent requirements of fuzzy mathematics.The heritability and the productive property of the degrees of separation axioms,and their relations are discussed.Secondly,we introduce the degree to which an(L,M)-fuzzy convex space is S0,S1 and S2 separated.Each(L,M)fuzzy convex space can be regarded to be S0,S1 and S2 separated to some degree.The heritability and the productive property of degree of separation axioms,and their relations are discussed.Finally,we introduce the degrees to which a function is convex to convex,convexity preserving and isomorphic in(L,M)fuzzy convex space.Their relationships with the degree of S0,S1 and S2 separation axioms in(L,M)-fuzzy convex space are discussed. |
