A number of completions have been applied to p.o.-groups -the Dedekind-Macneille completion of archimedean l.o. groups; the lateral completion of l.o. groups (Conrad [2]); and the orthocompletion of l.o. groups (Bernau [1]). Fuchs in [3] has considered a completion of p.o. groups having a non-trivial open interval topology -the only l.o. groups of this form being fully ordered. He applies an ordering, which arises from the original partial order, to the group of round Cauchy filters over this

more »

... pology; Kowalsky in [6] has shown that group, imbued with a suitable topology, is in fact the topological completion of the original group under its open interval topology. In this paper a slightly different ordering, also arising from the original order, is proposed for the group of round Cauchy filters; Fuchs' ordering can be obtained from this one as the associated order. §1 introduced the necessary underlying concepts, whilst §2 describes a slight short cut to the results needed from Kowalsky. §3 brings in new ordering described above, and shows that it has some desirable properties-the open interval topology corresponding to it is in fact the topology of the topological completion; the completion of the completion is o-isomorphic to the completion; and the completion is the ubique maximal extension of the original p.o. group in which this latter is sub-dense. In §4 the connection with Fuchs' completion is established, and it is noted that the tight Riesz property is preserved by completion. In conclusion, I should like to thank Professor J. B. Miller for his kind comments and encouragement, both of which have contributed to the compilation of this paper.