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Convergence with a regulator in lattice-ordered groups. (English) Zbl 1150.06020
Let $$G$$ be an archimedean ordered group and $$0 < u\in G$$. A sequence $$(x_n)$$ in $$G$$ is said to $$u$$-converge to an element $$x\in G$$ if for each $$p\in \mathbb N$$ there exists $$n_0\in \mathbb N$$ such that $$p| x_n-x| \leq u$$ for each $$n\in \mathbb N$$, $$n\geq n_0$$. The element $$u$$ is called a regulator of the convergence under consideration. It is proved that $$u$$-limits in $$G$$ are uniquely determined. The Cauchy completion of $$G$$ with respect to $$(u)$$-convergence is defined in a standard way. The main results of the paper are as follows: (i) If $$u$$ is a strong unit of $$G$$, then the Cauchy completion of $$G$$ does exist. (ii) If $$G_1$$ and $$G_2$$ are Cauchy completions of $$G$$, then there exists an isomorphisms $$\varphi$$ of $$G_1$$ onto $$G_2$$ leaving all elements of $$G$$ fixed.

##### MSC:
 06F15 Ordered groups