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Quotient groups and realization of tight Riesz groups

  • John Boris Miller (a1)

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Let (G, ≼) be an l-group having a compatible tight Riesz order ≦ with open-interval topology U, and H a normal subgroup. The first part of the paper concerns the question: Under what conditions on H is the structure of (G, ≼, ∧, ∨, ≦, U) carried over satisfactorily to by the canonical homomorphism; and its answer (Theorem 8°): H should be an l-ideal of (G, ≼) closed and not open in (G, U). Such a normal subgroup is here called a tangent. An essential step is to show that ≼′ is the associated order of ≦′.

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References

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[1]Cameron, Neil & Miller, J. B., Topology and axioms of interpolation in partially ordered spaces (J. für die reine u. angew. Math., to appear).
[2]Fuchs, L., Partially ordered algebraic systems (Pergamon, Oxford, 1963).
[3]Fuchs, L., ‘Riesz groups’, Ann. Scuola Norm. Sup. Pisa 19 (1965), 134.
[4]Loy, R. J. & Miller, J. B., ‘Tight Riesz groups’, J. Austral. Math. Soc. 13 (1972), 224240.
[5]Miller, J. B., ‘Higher derivations on Banach algebras’, Amer. J. Math. 92 (1970), 301331.
[6]Miller, J. B., ‘Tight Riesz groups and the Stone-Weierstrass theorem’ (Preprint, Monash University, 1970).
[7]Reilly, N. R., Compatible tight Riesz orders and prime subgroups. (Preprint, Simon Fraser University, 1971).
[8]Ribenboim, P., Théorie des groupes ordonnés (Universidad Nacional del Sur, Bahia Blanca, 1959).
[9]Speed, T. P. & Strzelecki, E., ‘A note on commutative l-groups’, J. Austral. Math. Soc. 12 (1971), 6974.
[10]Spirason, G. T. & Strzelecki, E., ‘A note on Pt-ideals’, J. Austral. Math. Soc. 14 (1972), 304310.
[11]Wirth, A., ‘Compatible tight Riesz orders’, J. Austral. Math. Soc. 15 (1973), 105111.
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