Some pathology involving pseudo -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 , two elements and are *pseudo-disjoint* if and either one is zero, or both are strictly positive and each -ideal which is maximal with respect to not containing contains , and vice versa. is a *pseudo lattice-group* if every element of can be written as a difference of pseudo-disjoint elements.

We prove the following theorem: suppose is an abelian pseudo lattice-group; if there is an and a finite set of pairwise pseudo-disjoint elements all of which exceed , and in addition this set is maximal with respect to the above properties, then is not a group of divisibility.

The main consequence of this result is that every so-called ``-group'' for a given partially ordered set , and where is a subgroup of the additive reals in their usual order, is a group of divisibility only if is a root system, and hence 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 whose group of divisibility is a lattice-group, the group of divisibility of the polynomial ring in one variable.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1973-0319825-X

Keywords:
Group of divisibility,
semivaluation,
pseudo-disjointness,
pseudo -group,
-group ,
primitive polynomial

Article copyright:
© Copyright 1973
American Mathematical Society