UNIFORM IDEÁL COMPLETIONS

The completion of a poset P by ideals (in the sense of Frink) is characterized abstractly as the smallest algebraic complete lattice containing P as a set of compact elements, while the completion by directed ideals is the smallest algebraic poset with the corresponding propert y It is known that both types of ideal completions may also be interpreted as certain topological completions. We show that a suitable intermediate completion by so-called Cauchy ideals may be regarded as a uniform completion of P. For lattices, the Cauchy ideal completion coincides with the Frink ideal completion (and with the completion by directed ideals, provided a least element exists). In the theory of partially ordered sets (posets), various types of ideals have been introduced for concrete constructions of certain universal completions. Prominent examples are AP, the Alexandroff completion by order ideals (lower sets), MP, the Dedekind-MacNeille completion by normal ideals (cuts),