The lattice of 𝑙-group varieties

Smith, J. E.

doi:10.1090/S0002-9947-1980-0552262-X

The lattice of $l$-group varieties
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by J. E. Smith

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

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

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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 ${\mathcal {S}_p}$ (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 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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