Scott algebraic closure space and its applications

Abstract

In this paper, we want to use the idea of fuzzy topological space generated by classical topological space to study generated algebraic closure space and its applications. Firstly, we show that the generated $\text{Q}$-Scott open set monad induced by classical Scott open set monad is a submonad of $\text{Q}$-Scott open set monad if and only if the underlying partial order of the quantale $\text{Q}$ is a frame. Then, we give a reasonable explanation for constructing Scott algebraic closure system on a frame $\text{Q}$ since it plays a similar role as Scott topology in topology. Finally, by the Scott algebraic closure system, we construct a monad on the category of $T_0$ separated algebraic closure spaces and show that the Eilenberg-Moore algebras of this monad are precisely frames.