A uniform approach to completions of posets

Abstract For a subset system Z and a subset selection Γ of Z−sets, we introduce two special types of subsets of a poset, called Δ Γ − continuously ⋁ − existing subsets and Δ Γ − closed subsets. We define the Z Γ − completion Z Γ ( P ) to be the set of all Δ Γ − continuously ⋁ − existing, Δ Γ − closed subsets of a given poset P. We prove that if Z is subset-hereditary, then (1) the Z Γ − completion Z Γ ( P ) is the smallest Z−complete subposet containing all principal ideals in the set of all Δ Γ − closed subsets of P; (2) any Δ Γ − continuous function f : P → L mapping into a Z−complete poset L extends uniquely to a Z ∨ − continuous function from the completion Z Γ ( P ) to L. The Z Γ − completion includes numerous special cases: the Dedekind-MacNeille completion, the Frink ideal completion, the ideal completion, the Z−completion, the Hoare powerdomain, etc.