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Pairs of partially ordered groups with the same convex subgroups. (English) Zbl 0615.06005
From the author’s introduction: ”J. Jakubík and M. Kolibiar [Czech. Math. J. 4(79), 1-27 (1954; Zbl 0059.026)] investigated pairs of distributive lattices $$L_ 1$$ and $$L_ 2$$ with the same underlying set such that the system of all convex sublattices of $$L_ 1$$ coincides with the system of all convex sublattices of $$L_ 2$$. They proved that $$L_ 1$$ and $$L_ 2$$ can differ only by duality of a direct factor.
The present paper is a contribution to the investigation of an analogous question concerning partially ordered groups. There are studied pairs of isolated abelian partially ordered groups (H,$$\leq)$$, (H,$$\leq ')$$ with the same underlying set and the same group operation such that the system of all convex subgroups of (H,$$\leq)$$ coincides with the system of all convex subgroups of (H,$$\leq ')$$. It is shown that instead of direct factors (as in the case examined in the above mentioned paper) we have now to deal with certain subdirect factors of (H,$$\leq)$$ and (H,$$\leq ')$$, respectively, which are either linearly ordered or trivially ordered.”
Reviewer: J.Jakubík
##### MSC:
 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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##### References:
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