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Splitting properties in Archimedean l-groups

Published online by Cambridge University Press:  09 April 2009

Marlow Anderson ,
Paul Conrad  and
Otis Kenny
Marlow Anderson
Affiliation:
The University of Kansas, Lawrence, Kansas 66045, U.S.A.
Paul Conrad
Affiliation:
The University of Kansas, Lawrence, Kansas 66045, U.S.A.
Otis Kenny
Affiliation:
The University of Kansas, Lawrence, Kansas 66045, U.S.A.
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Throughout this paper an l-group will always mean an archimedean lattice-ordered group and we shall confine our attention to such groups. An l-group splits if it is a cardinal summand of each l-group that contains it as an l-ideal. Suppose that G is an l-subgroup of an l-group H. Then G is large in H or H is an essential extension of G if for each l-ideal L≠0 of H, L∩G≠0. G is essentially closed if it does not admit any proper essential extension. Conrad (1971) proved that each essentially closed l-group splits, but not conversely.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 23 , Issue 2 , March 1977 , pp. 247 - 256
Copyright
Copyright © Australian Mathematical Society 1977

References

Bernau, S. (1966), ‘Orthocompletion of lattice groups’, Proc. London Math. Soc. 16, 107–130.CrossRefGoogle Scholar
Bernau, S. (1976), ‘Lateral and Dedekind completion of archimedean lattice groups’, J. London Math. Soc. 12, 320–322.CrossRefGoogle Scholar
Conrad, P. (1971), ‘The essential closure of an archimedean lattice-ordered group’, Duke Math. J. 38, 151–160.CrossRefGoogle Scholar
Conrad, P. (1973), ‘The hulls of representable l-groups and f-rings’, J. Austral. Math. Soc. 16, 385–415.CrossRefGoogle Scholar
Conrad, P. and McAlister, D. (1969), ‘The completion of a lattice-ordered group’, J. Austral. Math. Soc. 9, 182–208.CrossRefGoogle Scholar
Jakubik, J. (1963), ‘Representations and extensions of l-groups’, Czech. Math. J. 13, 267–283.CrossRefGoogle Scholar
Jakubik, J. (1974), ‘Splitting properties of lattice-ordered groups’, Czech. Math. J. 24, 259–271.CrossRefGoogle Scholar
McNeille, H. (1937), ‘Partially ordered sets’, Trans. Amer. Math. Soc. 42, 416–460.CrossRefGoogle Scholar
Pinsker, A. (1949), ‘Extended semi-ordered groups and spaces’, Uchen, Japiski Leningrad Cos. Ped. Inst. 86, 285–315.Google Scholar
Veksler, A. and Geiler, V. (1972), ‘Order and disjoint completeness of linear partially ordered spaces’, Sibirsk. Mat. Z. 13, 43–51.Google Scholar