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Some Krein-Milman theorems for order-convexity

Published online by Cambridge University Press:  09 April 2009

Andrew Wirth
Andrew Wirth
Affiliation:
Monash University, Clayton, Victoria 3168, Australia
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Analogues of the Krein-Milman theorem for order-convexity have been studied by several authors. Franklin [2] has proved a set-theoretic result, while Baker [1] has proved the theorem for posets with the Frink interval topology. We prove two Krein-Milman results on a large class of posets, with the open-interval topology, one for the original order and one for the associated preorder. This class of posets includes all pogroups. Cellular-internity defined in Rn by Miller [3] leads to another notion of convexity, cell-convexity. We generalize the definition of cell-convexity to abelian l-groups and prove a Krein-Milman theorem in terms of it for divisible abelian l-groups.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 3 , November 1974 , pp. 257 - 261
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Baker, K. A., ‘A Krein-Milman theorem for partially ordered sets’, Amer. Math. Monthly 76 (1969), 282–238.CrossRefGoogle Scholar
[2]Franklin, S. P., ‘Some results on order-convexity’, Amer. Math. Monthly 69 (1962), 357–359.CrossRefGoogle Scholar
[3]Millei, J. B., ‘Aczél's uniquness theorem and cellular internity’, Aequations Math. 5 (1970), 319–325.CrossRefGoogle Scholar
[4]Miller, J. B. and Cameron, N., ‘Topology and axioms of interpolation in partially ordered spaces’, J. für die reine u. angewandte Math. (to appear).Google Scholar
[5]Wirth, A., ‘Compatible tight Riesz orders’, J. Austral. Math. Soc. 15 (1973), 105–111.CrossRefGoogle Scholar