Definition. Endofunctors on \textbf {Cpo}^\textbf {R} [dt-005D]

An endofunctor on the category \textbf {Cpo}^\textbf {R} is a functor \textbf {Cpo}^\textbf {R} \rightarrow \textbf {Cpo}^\textbf {R}, i.e. a mapping \mathcal {F} on cpos together with a mapping \mathcal {F} on retraction pairs, such that:

1. If (f,g) is a retraction pair A \xtofrom [g]{f} B then \mathcal {F}(f,g) is a retraction pair \mathcal {F}(A)\xtofrom []{} \mathcal {F}(B)
2. \mathcal {F}(\mathsf {id}_A : A\xtofrom []{} A) = \mathsf {id}_{\mathcal {F}(A)} : \mathcal {F}(A)\xtofrom []{} \mathcal {F}(A)
3. \mathcal {F}(f \circ g) = \mathcal {F}(f) \circ \mathcal {F}(g)
(where \circ is composition of retraction pairs)