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Generalized intervals in partially ordered groups

Published online by Cambridge University Press:  24 October 2008

D. C. J. Burgess
D. C. J. Burgess
Affiliation:
Queen's UniversityBelfast
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Extract

1. Introduction. The present paper is chiefly concerned with a generalization, to be known as a ‘D-interval’, of the notion of interval or segment in an arbitrary partially ordered group. This idea is originally due to Duthie (2), but was developed by him only in a lattice. In analogy with the use of the interval in the normal sense, notions of ‘D-distributivity’ and ‘D-modularity’ are defined in terms of the D-interval, and analogues of known properties of lattice-groups or ‘l– groups’ can be formulated which might be valid when a lattice structure is no longer assumed to exist; in particular, an attempt is made to provide such a generalization of the result of Freudenthal (3) that every Z-group is a distributive lattice, but, for an arbitrary partially ordered group, it is shown that only an ‘approximation’ (in terms of non-Archimedean elements) to the desired result actually holds, although any Archimedean partially ordered group is necessarily D-distributive.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 55 , Issue 2 , April 1959 , pp. 165 - 171
Copyright
Copyright © Cambridge Philosophical Society 1959

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References

REFERENCES

(1)Birkhoff, G.Lattice theory, 2nd ed. (New York, 1948).Google Scholar
(2)Duthie, W. D.Segments of ordered sets. Trans. Amer. Math. Soc. 51 (1942), 1–14.CrossRefGoogle Scholar
(3)Freudenthal, H.Teilweise geordnete Moduln. Proc. K. Akad. Wet. Amst. 39 (1936), 641–51.Google Scholar
(4)Fuchs, L.Absolutes in partially ordered groups. Nederl. Akad. Wetensch. Proc. 52, 251–55; = Indagationes Math. 11 (1949), 66–70.Google Scholar
(5)Fuchs, L.On partially ordered groups. Nederl. Akad. Wetensch. Proc. 53, 828–34; = Indagationes Math. 12 (1950), 272–78.Google Scholar
(6)Loonstra, F.The class of partially ordered groups. Compositio Math. 9 (1951), 130–40.Google Scholar
(7)Shimbireva, . On the theory of partially ordered groups. Rec. Math. [Mat. Sbornik], N.S., 20 (1947), 145–78.Google Scholar