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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
##### MSC:
 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
##### Keywords:
abelian po-groups; saturated chains
##### References:
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