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Convergence with a fixed regulator in Archimedean lattice ordered groups. (English) Zbl 1150.06019
Let \(G\) be an Archimedean lattice-ordered group, \(0<u\in G\). The author works with the notions of \(u\)-convergent and \(u\)-fundamental sequences, which were introduced before. \(G\) is said to be \(u\)-Cauchy complete if every \(u\)-fundamental sequence in \(G\) is \(u\)-convergent in \(G\). A \(u\)-Cauchy completion of \(G\) is defined as an Archimedean \(u\)-Cauchy complete lattice-ordered group \(H\) containing \(G\) as an \(l\)-subgroup such that each element of \(H\) is a \(u\)-limit of a sequence in \(G\).
Another well-known type of completions of lattice-ordered groups is the Dedekind completion. Every Archimedean lattice-ordered group \(G\) admits a unique Dedekind completion \(G^{\wedge }\). In the present paper, there is proved that \(G^{\wedge }\) is Archimedean and \(u\)-Cauchy complete. Further, there is shown that \(G\) has a \(u\)-Cauchy completion \(G^{*}\), which can be considered as an \(l\)-subgroup of \(G^{\wedge }\). An example shows that \(G^{*}\neq G^{\wedge }\) in general.
Some further results:
If \(G\) is a direct product of lattice-ordered groups \(G_i\) \((i\in I)\), then \(G^{*}\) is isomorphic to the direct product of \(G^{*}_i\).
The ordered system of all \(u\)-Cauchy complete \(l\)-ideals of \(G\) containing \(u\) has a greatest element.

06F15 Ordered groups
20F60 Ordered groups (group-theoretic aspects)
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