Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-30T15:19:04.666Z Has data issue: false hasContentIssue false
  • English
  • Français

The Completion of a Lattice Ordered Group

Published online by Cambridge University Press:  09 April 2009

Paul Conrad  and
Donald McAlister
Paul Conrad
Affiliation:
Tulane University, New Orleans Queen's University, Belfast
Donald McAlister
Affiliation:
Tulane University, New Orleans Queen's University, Belfast
Rights & Permissions [Opens in a new window]

Extract

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.

A lattice ordered group(‘l-group’) is called complete if each set of elements that is bounded above has a least upper bound (and dually). A complete l-group is archimedean and hence abelian, and each archimedean l-group has a completion in the sense of the following theorem.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 9 , Issue 1-2 , February 1969 , pp. 182 - 208
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Banaschewski, B., ‘On lattice ordered groups’. Fundamenta Math. 55 (1964) 113–123.CrossRefGoogle Scholar
[2]Bernau, S. J., ‘Unique representation of archimedean lattice groups and normal archimedean lattice rings’. Proc. London Math. Soc. 15 (1965), 599–631.CrossRefGoogle Scholar
[3]Bernau, S. J., ‘Orthocompletion of lattice groups’. Proc. London Math. Soc. 16 (1966), 107–130.CrossRefGoogle Scholar
[4]Birkhoff, G., ‘Lattice theory’. American Math. Soc. Colloquium Pub. 25, (1948).Google Scholar
[5]Byrd, R., Tulane Dissertation (1966).Google Scholar
[6]Byrd, R., ‘Complete distributivity in lattice ordered groups’. Pacific J. Math. 20 (1967), 423–432.CrossRefGoogle Scholar
[7]Conrad, P., ‘Some structure theorems for lattice ordered groups’. Trans. American Math. Soc. 99 (1961), 212–240.CrossRefGoogle Scholar
[8]Conrad, P., ‘The relationship between the radical of a lattice-ordered group and complete distributivity’. Pacific J. Math. 14 (1964), 493–499.CrossRefGoogle Scholar
[9]Conrad, P., ‘The lattice of all convex l-subgroups of a lattice ordered group’. Czech. Math. J. 15 (1965), 101–123.CrossRefGoogle Scholar
[10]Conrad, P., ‘Archimedean extensions of lattice ordered groups’. J. Indian Math. Soc. 30 (1966), 131–160.Google Scholar
[11]Conrad, P., ‘Lateral completions of lattice ordered groups’. (To appear). Proc. London Math. Soc.Google Scholar
[12]Fuchs, L., Partially Ordered Algebraic Systems. (Pergamon Press. 1963).Google Scholar
[13]Iwasawa, K., ‘On the structure of conditionally complete lattice-groups’. Japan J. Math. 18 (1943), 777–789.CrossRefGoogle Scholar
[14]Jaffard, P., ‘Sur le spectre d'un groupe réticule et l'unicité des réalisations irréducibles’. Ann. Univ. Lyon, 22 (1950), 43–47.Google Scholar
[15]Jakubik, J., ‘Representations and extensions of l-groups’. Czech. Math. J. 13 (1963), 267–283.CrossRefGoogle Scholar
[16]Johnson, D. G. and Kist, J. E., ‘Complemented ideals and extremely disconnected spaces’. Archiv der Math. 12 (1961), 349–354.CrossRefGoogle Scholar
[17]Johnson, D. G. and Kist, J. E., ‘Prime ideals in vector lattices’. Canadian J. Math. 14 (1962), 512–528.CrossRefGoogle Scholar
[18]Lorenz, K., ‘Über Strukturverbande von Verbandsgruppen’. Acta. Math. Soc. Hungary. 13 (1962), 55–67.CrossRefGoogle Scholar
[19]MacNeille, H., ‘Partially ordered sets’. Trans American Math. Soc. 42 (1937), 416–460.CrossRefGoogle Scholar
[20]Topping, D., ‘Some homological pathology in vector lattices. Canadian J. Math. 17 (1963), 411–428.CrossRefGoogle Scholar
[21]Weinberg, E., ‘Completely distributive lattice ordered groups’. Pacific J. Math. 12 (1962), 1131–1137.Google Scholar
[22]Weinberg, E., ‘Free lattice ordered abelian groups’. Math. Annalen 151 (1963), 187–189.CrossRefGoogle Scholar
[23]Weinberg, E., ‘Free lattice ordered abelian groups, II’. Math. Annalen 159 (1965), 217–222.CrossRefGoogle Scholar