Quantale-valued convex structures as lax algebras

Based on a unital and commutative quantale (Q,⁎), a Q-valued lax extension of the nonempty finite powerset monad and a Q-valued finitary closure space (also called algebraic Q-valued closure space) are introduced. It is proved that the category of (Pf,Q)-categories with respect to the Q-valued lax extension of the nonempty finite powerset monad Pf is isomorphic to that of Q-valued finitary closure spaces. Considering the Q-valued finitary closure spaces as linkages, it is shown that balanced Q-convex structures can be treated as (Pf,Q)-categories when Q is required to be a frame and Q-fuzzifying convex structures can be treated as (Pf,Q)-categories when Q is required to be a completely distributive De Morgan algebra.