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Cantor extension of a half lattice ordered group. (English) Zbl 0938.06014
Half lattice ordered groups have been studied by M. Giraudet and F. Lucas [Fundam. Math. 139, No. 2, 75-89 (1991; Zbl 0766.06014)]. Let \(G\) be a half lattice ordered group; then there are subsets \(G{\uparrow}\) and \(G{\downarrow}\) of \(G\) such that \(G{\uparrow}\cup G{\downarrow}=G\) and \(G{\uparrow}\cap G{\downarrow}=\emptyset \). Moreover, \(G{\uparrow}\) is a lattice ordered group (under the operations induced from \(G\)). The author deals with the case when the group operation in \(G{\uparrow}\) is abelian. The Cantor extension \(G'\) of \(G\) is defined by the requirements that a) \(G'\) is \(o\)-complete; b) \(G\) is an hl-subgroup of \(G'\); c) every element of \(G'\) is an \(o\)-limit in \(G'\) of a fundamental sequence in \(G\). It is proved that the Cantor extension of \(G\) exists and that it is uniquely determined up to isomorphism leaving all the elements of \(G\) fixed.

MSC:
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
06F15 Ordered groups
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References:
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