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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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