Strongly Finitary Monads and Continuous Algebras

A monad on the category $\mathsf{CPO}$ of complete posets is strongly finitary if it is an enriched left Kan extension of its restriction to finite discrete cpos. We prove that these monads correspond bijectively to varieties of continuous algebras. These are algebras acting on cpos such that operations are continuous. We also prove that in $\mathsf{CPO}$, in fact any cartesian closed category, directed colimits commute with finite products. We derive a characterization of strong finitarity as the preservation of directed colimits and reflexive coinserters.