The lattice of $l$group varieties
Author:
J. E. Smith
Journal:
Trans. Amer. Math. Soc. 257 (1980), 347357
MSC:
Primary 06F15; Secondary 06B20
DOI:
https://doi.org/10.1090/S0002994719800552262X
MathSciNet review:
552262
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Abstract  References  Similar Articles  Additional Information
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 latticeordered groups (lgroups). Martinez has shown that the set L of all lgroup varieties forms a lattice under set inclusion with a compatible associative multiplication. Certain varieties ${\mathcal {S}_p}$ (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 ${\mathcal {S}_n} (n \in N)$, and it is established that all infinite collections of the ${\mathcal {S}_n}$ have the same least upper bound in L. Martinez has also classified lgroups 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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Additional Information
Keywords:
Latticeordered group,
variety,
lattice,
Scrimger variety,
torsion class
Article copyright:
© Copyright 1980
American Mathematical Society