Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-26T10:22:49.387Z Has data issue: false hasContentIssue false
  • English
  • Français

The lateral completion of an arbitrary lattice group

Published online by Cambridge University Press:  09 April 2009

S. J. Bernau
S. J. Bernau
Affiliation:
University of Texas Austin, Texas 78712, U.S.A.
Rights & Permissions [Opens in a new window]

Summary

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper shows that every lattice group G can be densely embedded in a unique laterally complete lattice group H (the lateral completion of G). All reasonable structure properties of G are inherited by H and we have the following relationships between the ideal radical L(G), the distributive radical D(G) and the radical R(G) of G and the corresponding radicals of H. .

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 19 , Issue 3 , May 1975 , pp. 263 - 289
Copyright
Copyright © Australian Mathematical Society 1975

References

Amemiya, I. G. (1953), ‘A general spectral theory in semi-ordered linear spaces’, J. Fac. Sci. Hokkaido Univ. Ser. I, 12, 111–156.Google Scholar
Bernau, S. J. (1966), ‘Orthocompletion of lattice groups’, Proc. London Math. Soc. (3) 16, 107–130.CrossRefGoogle Scholar
Bernau, S. J. (1965), ‘Unique representation of Archimedean lattice groups and normal Archimedean lattice rings’, Proc. London Math. Soc. (3) 15, 599–631.CrossRefGoogle Scholar
Byrd, R. D. (1967), ‘Complete distributivity in lattice-ordered groups’, Pacific J. Math. 20, 423–432.CrossRefGoogle Scholar
Byrd, R. D. and Lloyd, J. T. (1969), ‘A note on lateral completions in lattice-ordered groups’, J. London Math. Soc. (2) 1, 358–362.CrossRefGoogle Scholar
Byrd, R. D. and Lloyd, J. T. (1967), ‘Closed subgroups and complete distributivity in lattice- ordered groups’, Math. Zeit. 101, 123–130.CrossRefGoogle Scholar
Conrad, P. F., ‘Lateral completion of lattice-ordered groups’, Proc. London Math. Soc. (3) 19, 444–480.Google Scholar
Conrad, P. F. (1964), ‘The relation between the radical of a lattice-orderd group and complete distributivity’, Pacific J. Math. 14, 493–499.CrossRefGoogle Scholar
Conrad, P. F. (1961), ‘Some structure theorems for lattice-ordered groups’, Trans. Amer. Math. Soc. 99, 212–240.CrossRefGoogle Scholar
Conrad, P. F. (1965), ‘The lattice of all conve 1-subgroups of a lattice-ordered group’, Czech. Math. J. 15, 101–123.CrossRefGoogle Scholar
Holland, C. (1963), ‘The lattice-ordered group of automorphisms of an ordered set’, Michigan Math. J. 10, 71–82.CrossRefGoogle Scholar
Lorenzen, P. (1949), ‘Uber halbgeordnete Gruppen’, Math. Zeit. 52 484–526.Google Scholar
Nakano, N. (1950), Modern Spectarl Theory, (Tokyo 1950).Google Scholar
Pinkser, A. G. (1949), ‘E tended semiordered groups and spaces’, Uchen Zapiski Lemingrad Gos. Ped. Inst. 86, 236–365.Google Scholar
Veksler, A. I. and Geiler, V. A. (1972), ‘Order and disjoint comppeteness of linear partially ordered spaces’, Siberian Math. J. 13, 30–35,CrossRefGoogle Scholar
(Translated from Sibirsk. Math. Z. 13 (1972), 43–51).Google Scholar