Is the lattice of torsion classes algebraic?

Author:
Jorge Martinez

Journal:
Proc. Amer. Math. Soc. **63** (1977), 9-14

MSC:
Primary 06A60

MathSciNet review:
0552652

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Abstract: The answer is yes, if ...

This note attempts to give amplification to the above statement, while at the same time arriving at a reasonable description of this lattice.

The main theorem of the paper is no doubt the assertion that the lattice of torsion classes of lattice-ordered groups is completely distributive. The proof of this theorem depends on the notion of a *value selector*, and should not. As a consequence of this, one obtains a (local) decomposition theorem which is canonical (in every sense of the word) and always works.

**[1]**P. Conrad,*Lattice-ordered groups*, Tulane Univ., 1970.**[2]**Paul Conrad,*Epi-archimedean groups*, Czechoslovak Math. J.**24 (99)**(1974), 192–218. MR**0347701****[3]**Jorge Martinez,*Unique factorization in partially ordered sets*, Proc. Amer. Math. Soc.**33**(1972), 213–220. MR**0292723**, 10.1090/S0002-9939-1972-0292723-5**[4]**Jorge Martinez,*Torsion theory for lattice-ordered groups*, Czechoslovak Math. J.**25(100)**(1975), 284–299. MR**0389705****[5]**Jorge Martinez,*Torsion theory for lattice-ordered groups. II. Homogeneous 𝑙-groups*, Czechoslovak Math. J.**26(101)**(1976), no. 1, 93–100 (English, with Loose Russian summary). MR**0389706**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1977-0552652-0

Keywords:
Torsion class,
torsion radical,
value selector,
algebraic lattice,
compact element

Article copyright:
© Copyright 1977
American Mathematical Society