The theory of abstract convexity exists in many mathematical branches such as algebra, topology and order. The fuzzification is one of important directions towards the discussion of abstract convexity. The theory of (L, M)-convex spaces is one very general fuzzy convexity since it includes many other fuzzy convexities as its special case. In this paper, we shall introduce a subcategory of (L, M)-convex spaces and study its property. This subcategory is called enriched (L, M)-convex spaces and denoted by ELMCS. The main results are: (1) ELMCS is a topological category; (2) ELMCS is a coreflective subcategory of the category of (L, M)-convex spaces; (3) the category of M-fuzzifying convex spaces (denoted by MCS) can be embedded in the category ELMCS as a reflective subcategory.
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