The order-K-ification monads

Monads prove to be useful mathematical tools in theoretical computer science, notably in denoting different effects of programming languages. In this paper, we investigate a type of monads which arise naturally from Keimel and Lawson’s $\mathbf{K}$-ification.

A subcategory of $\mathbf{TOP}_{\mathbf{0}}$ is called of type $\mathrm{K}^{*}$ if it consists of monotone convergence spaces and is of type $\mathrm K$ in the sense of Keimel and Lawson. Each such category induces a canonical monad $\mathcal K$ on the category $\mathbf{DCPO}$ of dcpos and Scott-continuous maps, which is called the order-$\mathbf{K}$-ification monad in this paper. First, for each category of type $\mathrm{K}^{*}$, we characterize the algebras of the corresponding monad $\mathcal K$ as k-complete posets and algebraic homomorphisms as k-continuous maps, from which we obtain that the order-$\mathbf{K}$-ification monad gives the free k-complete poset construction over the category $\mathbf{POS}_{\mathbf{d}}$ of posets and Scott-continuous maps. In addition, we show that all k-complete posets and Scott-continuous maps form a Cartesian closed category. Moreover, we consider the strongness of the order-K-ification monad and conclude with the fact that each order-K-ification monad is always commutative.