The lattice of -group varieties

Author:
J. E. Smith

Journal:
Trans. Amer. Math. Soc. **257** (1980), 347-357

MSC:
Primary 06F15; Secondary 06B20

MathSciNet review:
552262

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Abstract: For any type of abstract algebra, a variety is an equationally defined class of such algebras. Recently, attempts have been made to study varieties of lattice-ordered groups (*l*-groups). Martinez has shown that the set **L** of all *l*-group varieties forms a lattice under set inclusion with a compatible associative multiplication. Certain varieties (*p* prime) have been proved by Scrimger to be minimal nonabelian varieties in **L**. In the present paper, it is shown that these varieties can be used to produce varieties minimal with respect to properly containing various other varieties in **L**. Also discussed are the relations among the , and it is established that all infinite collections of the have the same least upper bound in **L**. Martinez has also classified *l*-groups using torsion classes, a generalization of the idea of varieties. It is proved here that **L** is not a sublattice of **T**, the lattice of torsion classes.

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DOI:
https://doi.org/10.1090/S0002-9947-1980-0552262-X

Keywords:
Lattice-ordered group,
variety,
lattice,
Scrimger variety,
torsion class

Article copyright:
© Copyright 1980
American Mathematical Society