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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.

06F15 Ordered groups