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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.

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