In this paper, we generalize a series of research work about convexity on classical partially ordered sets to fuzzy partially ordered sets (L-posets). Taking a complete Heyting algebra as the truth value structure, we propose an L-ordered L-convex structure on an L-poset and give its corresponding L-convex hulls. We characterize the L-ordered L-convex sets in terms of four kinds of cut sets of L-subsets, and discuss the product of L-ordered L-convex sets. We also discuss L-convexity-preserving (resp.,L-convex-to-convex) mappings. After that, with a consideration of the degree to which an L-subset is an L-ordered L-convex set, an L-ordered (L, L)-fuzzy convex structure is introduced. The properties such as equivalent descriptions, the product and (L, L)-fuzzy convexity-preserving mappings are analyzed.
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