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On partially ordered groups of locally finite length. (English) Zbl 0598.06009
For any cardinal \(\alpha\geq 2\) denote by \({\mathfrak G}_{\alpha}\) the class of all abelian po-groups G such that the poset (G,\(\leq)\) is directed, of locally finite length, all saturated chains from a to b \((a<b)\) have the same length and the subset of all elements in G covering zero has cardinality \(\alpha\). E.g., the direct sum \(D_{\alpha}\) of \(\alpha\) copies of \({\mathbb{Z}}(+)\) with the natural linear order belongs to \({\mathfrak G}_{\alpha}\). Let \(f=\alpha_ 1x_ 1+...+\alpha_ nx_ n\), \(\alpha_ i\in {\mathbb{Z}}\), be any nonzero element in the free abelian group \(F_{\alpha}\) with the set \(X=\{x_ i|\) \(i\in I\}\) of free generators. Putting \(f>0\) for elements \(f\in F_{\alpha}\) with all \(\alpha_ i>0\) we get a po-group \(F'_{\alpha}\) isomorphic to \(D_{\alpha}\in {\mathfrak G}_{\alpha}\). The author proves that for a po- group G to be in \({\mathfrak G}_{\alpha}\) is equivalent to be isomorphic with \(F'_{\alpha}/K\) where \(K\leq F'_{\alpha}\) satisfies the conditions: (i) if \(x_ 1\) and \(x_ 2\) are distinct elements of X then \(x_ 1-x_ 2\not\in K\), and (ii) \(K\leq Ker \psi\) where the mapping \(\psi\) : F\({}'_{\alpha}\to {\mathbb{Z}}\) is defined by the rule \(\psi (f)=\sum_{i}\alpha_ i\). It is proved also that for each cardinal \(\alpha\geq 2\) there exists an infinite set of non-isomorphic po-groups belonging to \({\mathfrak G}_{\alpha}\).
Reviewer: U.Kaljulaid
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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