Frame-valued Scott open set monad and its algebras

Abstract

For a frame Q, this paper studies the Kleisli monoids and Eilenberg-Moore algebras with respect to the Q-Scott open set monad. Firstly, we show that the Q-Scott open set functor can form a monad, and prove that the Kleisli monoids of this monad are exactly the algebraic Q-ordered closure spaces or equivalently the strong Q-convex spaces. Then we show that the Eilenberg-Moore algebras of Q-Scott open set monad are precisely the algebraic Q-modules or equivalently the fuzzy frames. Finally, we construct the strong Q-Scott open set monad, and study the corresponding Kleisli monoids and the Eilenberg-Moore algebras.