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Tight Riesz groups

Published online by Cambridge University Press:  09 April 2009

R. J. Loy  and
J. B. Miller
R. J. Loy
Affiliation:
Carleton University Ottawa
J. B. Miller
Affiliation:
Trent UniversityPeterborough
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The theory of partially ordered topological groups has received little attention in the literature, despite the accessibility and importance in analysis of the group Rm. One obstacle in the way of a general theory seems to be, that a convenient association between the ordering and the topology suggests that the cone of all strictly positive elements be open, i.e. that the topology be at least as strong as the open-interval topology U; but if the ordering is a lattice ordering and not a full ordering then U itself is already discrete. So to obtain in this context something more interesting topologically than the discrete topology and orderwise than the full order, one must forego orderings which make lattice-ordered groups: in fact, the partially ordered group must be an antilattice, that is, must admit no nontrivial meets or joins (see § 2, 10°).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 13 , Issue 2 , March 1972 , pp. 224 - 240
Copyright
Copyright © Australian Mathematical Society 1972

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