Generalized Continuous Posets and a New Cartesian Closed Category

Abstract

With every subset selection $\mathcal {Z}$ for posets, there is associated a certain ideal completion $\mathcal {Z}^\triangle$. As shown by Erné, such completions help to extend classical results on domains and similar structures in the absence of the required joins. Some results about $\mathcal {Z}$–predistributive or $\mathcal {Z}$–precontinuous posets and $\mathcal {Z}^\triangle$–continuous functions are summarized and supplemented. In particular, several central results on function spaces in domain theory are extended to the setting of productive closed subset selections. The category FSBP, in which objects are finitely separated and upper bounded posets and arrows are $\mathcal D^{\triangle}-$continuous functions between them, is shown to be cartesian closed.