Abstract
We define the $ \scr A $ -ideals of a poset – or equally of a quasi-ordered set – for various collections of subsets $ \scr A $ and corresponding $ \scr A $ -ideal continuity for functions. This leads us to a choice-free $ \scr A $ -ideal continuous imbedding of a poset into a $ \scr B $ -join complete poset with an appropriate universal mapping property. Topological applications include the imbedding of Scott spaces and Alexandrov spaces into up-complete Scott spaces.
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