Convex structures via convex L-subgroups of an L-ordered group

In this paper, we first characterize the convex L-subgroup of an L-ordered group by means of four kinds of cut sets of an L-subset. Then we consider the homomorphic preimages and the product of convex L-subgroups. After that, we introduce an L-convex structure constructed by convex L-subgroups. Furthermore, the notion of the degree to which an L-subset of an L-ordered group is a convex L-subgroup is proposed and characterized. An L-fuzzy convex structure which results from convex L-subgroup degree is imported naturally, and its L-fuzzy convexity preserving mappings investigated.