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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Some pathology involving pseudo $ l$-groups as groups of divisibility


Author: Jorge Martinez
Journal: Proc. Amer. Math. Soc. 40 (1973), 333-340
MSC: Primary 06A55
DOI: https://doi.org/10.1090/S0002-9939-1973-0319825-X
MathSciNet review: 0319825
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Abstract: In a partially ordered abelian group $ G$, two elements $ a$ and $ b$ are pseudo-disjoint if $ a,b \geqq 0$ and either one is zero, or both are strictly positive and each $ o$-ideal which is maximal with respect to not containing $ a$ contains $ b$, and vice versa. $ G$ is a pseudo lattice-group if every element of $ G$ can be written as a difference of pseudo-disjoint elements.

We prove the following theorem: suppose $ G$ is an abelian pseudo lattice-group; if there is an $ x > 0$ and a finite set of pairwise pseudo-disjoint elements $ {x_1},{x_2}, \cdots ,{x_k}$ all of which exceed $ x$, and in addition this set is maximal with respect to the above properties, then $ G$ is not a group of divisibility.

The main consequence of this result is that every so-called ``$ v$-group'' $ V(\Lambda ,{R_\lambda })$ for a given partially ordered set $ \Lambda $, and where $ {R_\lambda }$ is a subgroup of the additive reals in their usual order, is a group of divisibility only if $ \Lambda $ is a root system, and hence $ V(\Lambda ,{R_\lambda })$ is a lattice-ordered group. We do give examples of pseudo lattice-groups which are not lattice-groups, and yet are groups of divisibility.

Finally, we compute for each integral domain $ D$ whose group of divisibility is a lattice-group, the group of divisibility of the polynomial ring $ D[x]$ in one variable.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0319825-X
Keywords: Group of divisibility, semivaluation, pseudo-disjointness, pseudo $ l$-group, $ v$-group $ V(\Lambda ,{R_\lambda })$, primitive polynomial
Article copyright: © Copyright 1973 American Mathematical Society