Vol. 15, No. 4, 1965

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Some considerations on convergence in abelian lattice-groups

Fredos Papangelou

Vol. 15 (1965), No. 4, 1347–1364

We define α-convergence in an abelian l-group as follows: The net (xi)iI α-converges to x if x is the only element such that x = ii0(xi x) = ii0(xi x) for every i0 I. In an Archimedean l-group (xi) α-converges to x if and only if for every a and b the net (a xi) b order-converges (in the ordinary sense) to (a x) b. In general α-convergence is weaker than this latter condition and is considerably more natural in the non-Archimedean case. The algebraic operations of an arbitrary abelian l-group G are continuous relative to α-convergence. If G is completely distributive its α-convergence derives from a Hausdorff group-topology. Three sufficient conditions are given for the preservation of the α-convergence of an l-group G when it is embedded in another l-group E. In an appendix, we formulate a necessary and sufficient condition in order that an abstract sequential convergence derive from a topology.

Mathematical Subject Classification
Primary: 06.78
Secondary: 20.00
Received: 8 October 1964
Published: 1 December 1965
Fredos Papangelou