Archimedean-like classes of lattice-ordered groups

Author:
Jorge Martinez

Journal:
Trans. Amer. Math. Soc. **186** (1973), 33-49

MSC:
Primary 06A55

MathSciNet review:
0332614

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Abstract: Suppose denotes a class of totally ordered groups closed under taking subgroups and quotients by *o*-homomorphisms. We study the following classes: (1) , the class of all lattice-ordered groups which are subdirect products of groups in ; (2) , the class of lattice-ordered groups in having all their *l*-homomorphic images in ; Para , the class of lattice-ordered groups having all their principal convex *l*-subgroups in . If is the class of archimedean totally ordered groups then Para is the class of archimedean lattice-ordered groups, is the class of subdirect products of reals, and consists of all the hyper archimedean lattice-ordered groups.

We show that under an extra (mild) hypothesis, any given representable lattice-ordered group has a unique largest convex *l*-subgroup in ; this socalled hyper- -kernel is a characteristic subgroup. We consider several examples, and investigate properties of the hyper- -kernels.

For any class as above we show that the free lattice-ordered group on a set *X* in the variety generated by is always in . We also prove that has free products.

**[1]**R. Bleier,*Dissertation*, Tulane University, New Orleans, La., 1971.**[2]**P. M. Cohn,*Universal algebra*, Harper & Row, Publishers, New York-London, 1965. MR**0175948****[3]**Paul Conrad,*A characterization of lattice-ordered groups by their convex𝑙-subgroups*, J. Austral. Math. Soc.**7**(1967), 145–159. MR**0214521****[4]**-,*Lattice-ordered groups*, Tulane University, New Orleans, La., 1970.**[5]**Paul Conrad and Donald McAlister,*The completion of a lattice ordered group*, J. Austral. Math. Soc.**9**(1969), 182–208. MR**0249340****[6]**L. Fuchs,*Partially ordered algebraic systems*, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1963. MR**0171864****[7]**Jorge Martinez,*Free products of abelian 𝑙-groups*, Czechoslovak Math. J.**23(98)**(1973), 349–361. MR**0371770****[8]**Hanna Neumann,*Varieties of groups*, Springer-Verlag New York, Inc., New York, 1967. MR**0215899****[9]**Elliot Carl Weinberg,*Free lattice-ordered abelian groups*, Math. Ann.**151**(1963), 187–199. MR**0153759****[10]**Samuel Wolfenstein,*Valeurs normales dans un groupe réticulé*, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)**44**(1968), 337–342 (French, with Italian summary). MR**0234887**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1973-0332614-X

Keywords:
Closed class of *o*-groups,
residually- *l*-groups,
hyper- *l*-groups,
para- *l*-groups,
hyper- kernel,
*c*-archimedean *l*-groups,
radical

Article copyright:
© Copyright 1973
American Mathematical Society