The order-<bold>K</bold>-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 K-ification.A subcategory of TOP0 is called of type K-& lowast; if it consists of monotone convergence spaces and is of type KK in the sense of Keimel and Lawson. Each such category induces a canonical monad K on the category DCPO of dcpos and Scott-continuous maps, which is called the order-K-ification monad in this paper. First, for each category of type K-& lowast;, we characterize the algebras of the corresponding monad K as k-complete posets and algebraic homomorphisms as k-continuous maps, from which we obtain that the order-K-ification monad gives the free k-complete poset construction over the category POSd 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.