Extensions of Ordered Groups and Sequence Completion
Charles W. Holland · Transactions of the American Mathematical Society · 1963
Introduction.In this paper the relationships between certain extensions of totally (linearly) ordered groups (o-groups) are studied.The additive notation is employed although the groups are not, in general, assumed to be abelian.The four types of extensions considered are defined as follows :Let G S H be o-groups.(a) H is an a-extension of G if for every 0 < he H there exists ge G and a positive integer n such that h ^ g ^ nh.(b) H is a b-extension of G if for every 0 < n e H there exists g e G and a positive integer n such that g ^ n :£ ng.(c) H is a c-extension of G if for every 0 < h e Ü there exists g e G such that for all integers n, n(h -g) < h.(t) H is a t-extension of G if for every h,h',h"eH with h < h' < h" there exists g e G such that h < g < h".If xe {a, b,c,f}, G is x-closed if G has no proper x-extensions.The connecting notion between the different types of extensions is that of sequences in ogroups; of particular importance are cauchy sequences (definition in §2) and pseudo sequences (definition in §3).Several authors have discussed similar concepts, but in different or more special cases.Cohen and Goffman [3 ; 4] consider cauchy sequences in abelian groups; Everett and Ulam [8] consider countable cauchy sequences; Banaschewski [1] deals with cauchy filters in partially ordered groups; Gravett [9] relates pseudo sequences to c-extensions in divisible abelian groups.§2 contains definitions and a basic lemma concerning sequences.It is also shown that every r-extension and every c-extension is a ¿-extension, and that every b-extension is an a-extension.Some results concerning r-extensions and cauchy sequences ("j" theorems) are stated without proof.In §3 some theorems ("C" theorems) concerning pseudo sequences and c-extensions, including analogues to the "T" theorems, are proved.In §4 are stated the "B" theorems concerning sequences and b-extensions with proofs only sketched, as the proofs