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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**Paul Conrad,*Representation of partially ordered abelian groups as groups of real valued functions*, Acta Math.**116**(1966), 199–221. MR**201536****[2]**-,*Lattice ordered groups*, Tulane University, New Orleans, La., 1970.**[3]**Paul Conrad and J. Roger Teller,*Abelian pseudo lattice ordered groups*, Publ. Math. Debrecen**17**(1970), 223–241 (1971). MR**323661****[4]**Paul Jaffard,*Comtribution à l’étude des groupes ordonnés*, J. Math. Pures Appl. (9)**32**(1953), 203–280 (French). MR**57869****[5]**-,*Les systèmes d'idéaux*, Travaux et Recherches Mathematiques, IV, Dunod, Paris, 1960. MR**22**#5628.**[6]**W. Krull,*Allgemeine Bewertungstheorie*, J. Reine Angew. Math.**167**(1931), 160-196.**[7]**Joe L. Mott,*The group of divisibility and its applications*, Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Springer, Berlin, 1973, pp. 194–208. Lecture Notes in Math., Vol. 311. MR**0337943****[8]**Jack Ohm,*Semi-valuations and groups of divisibility*, Canadian J. Math.**21**(1969), 576–591. MR**242819**, https://doi.org/10.4153/CJM-1969-065-9**[9]**Daniel Zelinsky,*Topological characterization of fields with valuations*, Bull. Amer. Math. Soc.**54**(1948), 1145–1150. MR**28303**, https://doi.org/10.1090/S0002-9904-1948-09141-8

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
06A55

Retrieve articles in all journals with MSC: 06A55

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