Completions of Partially Ordered Sets and Universal Algebra

The results' in this paper were motivated by a search for the answers to the following questions : When can a given partially ordered algebra be embedded in a complete partially ordered algebra in such a fashion that the original algebra is dense and that the operations acquire or maintain continuity properties? What type of identities are preserved under such embeddings? Properties of extensions and completions do not depend on the algebraic structure but only on the partial order. Thus in § § 2-4 we consider extensions and completions of partially ordered sets. We are led to the notions of order preserving mappings, order true mappings (2. 14), and abnormal extensions Z (2. 6) of a partially ordered set X. The analogue of the statement that a subset X of a topological space Z is dense in Z is the statement that Z is an abnormal extension of X. If oc is an X preserving order true mapping of an abnormal extension Z of X onto Y, then Y is an abnormal extension of oc(Y), and oc preserves suprema and infima (2.27). We construct a partially ordered set Z = {z: z EZ} of equivalence classes (2. 19) each class consisting of all elements ' which can possibly be identified under X preserving order true mappings. We show that Izl ~ 3, for all z E Z (2. 20 and 2. 22). If Y is an .if' preserving order true image of Z, then the natural mapping 'l' of Z onto Z, which is order true, factors trough Y (2. 35) and Y is isomorphic to Z (2. 36). In fact in the linearly ordered case Z is essentially the unique normal extension of X (2. 42). The structure of abnormal extensions is completely determined in the case of a linearly ordered set X (§ 4): A universal abnormal extension fF f (X) is constructed such that every other s' uch extension is faithfully embeddable into it (4. 14). The question of which extensions are completions is also solved in that it is shown that they are essentially homomorphic images of fF f(X) (4. 11). It is further shown that there is ,a minimal completion, the normal completion (CLIFFORD [1]) which is both a homomorphic image of all completions and isomorphically embeddable in all completions. In the case of partially ordered sets it is not known to us if a …