Bin Pang

Based on a completely distributive lattice L, we propose a degree approach to L-fuzzy ordered subsemigroups of an ordered semigroup. Firstly, we introduce the concept of L-fuzzy ordered subsemigroup degree function with respect to an ordered semigroup, which can be used to describe the degree to which an L-fuzzy subset of the ordered semigroup becomes an L-fuzzy ordered subsemigroup. Secondly, we use four kinds of cut sets depending on L to characterize the L-fuzzy ordered subsemigroup degree function. Finally, we provide a natural way to construct an L-fuzzy convex structure on an ordered semigroup via the L-fuzzy ordered subsemigroup degree function, and show that the homomorphism between two ordered semigroups is an L-fuzzy convexity-preserving mapping and the monohomomorphism is an L-fuzzy convex-to-convex mapping between the resulting L-fuzzy convex spaces.

Based on a complete Heyting algebra L, we first propose the concept of L-quasi-convex spaces and construct an adjunction between the category of L-S 0-quasi-convex spaces and the opposite category of complete L-ordered sets. Then we present the concept of weakly fuzzy algebraic lattices and prove that an L-quasi-convex structure endowed with the fuzzy inclusion order is precisely a weakly fuzzy algebraic lattice. Secondly, we introduce the notion of sobriety in Lquasi-convex spaces from the perspective of categorical equivalence, showing that the category of sober L-quasi-convex spaces is dually equivalent to that of weakly fuzzy algebraic lattices. Finally, we construct a monad on the category of L-S 0-quasi-convex spaces and obtain that the Eilenberg–Moore algebras of this m onad are precisely sober L-quasi-convex spaces.