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Convergences on lattice ordered groups with a finite number of disjoint elements. (English) Zbl 1141.06016
Convergence of sequences in totally ordered groups can be considered equivalently either (1) in terms of the interval topology, or (2) with relation to sequences of positive elements whose infimum is \(0\). Although both of these methods can be extended to lattice-ordered groups, their relationship is less exact and there are many possible generalizations which have been investigated during the last 50 years. For example, see [B. Banaschewski, Math. Nachr. 16, 51–71 (1957; Zbl 0077.03302)].
The paper under review is the 13th in a sequence by the present author and by M. Harminc, beginning in 1984 and dealing with convergence of sequences in lattice-ordered groups in the sense of (2). A congruence of a lattice-ordered group \(G\) is a set of sequences of positive elements of \(G\) which converge to \(0\) subject to a set of conditions described in the paper. These congruences are ordered by containment, which is related in a general sense to the rate of convergence. In the case of totally ordered groups, there are at most two congruences, namely those sequences which are finally \(0\), and those which are eventually bounded above by a monotone sequence whose infimum is \(0\). The author generalizes this observation to show that in a lattice-ordered group in which every set of disjoint positive elements is finite, the set of congruences is a finite Boolean algebra; and in the abelian case, the converse is also true.

MSC:
06F15 Ordered groups
Citations:
Zbl 0077.03302
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References:
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