Cancellation law and attainable classes of linear $\Omega$-algebras
Matematičeskie zametki, Tome 16 (1974) no. 3, pp. 467-478
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With the aid of mixed linear $\Omega$-algebras we prove a theorem to the effect that the cancellation law is satisfied in a groupoid of subvarieties of a variety of $\Omega$-algebras linear over a field and given by identities of zero order. We show that in some varieties of $\Omega$-algebras linear over an infinite ring of principal ideals there are no nontrivial finitely attainable subvarieties. As examples of such varieties we cite the varieties of all $\Omega$-rings, of all rings, of commutative or anticommutative rings ($\Omega$-rings), of Lie rings, et al. In the case of anticommutative rings ($\Omega$-rings) this property holds for $\Omega$-algebras, linear over an arbitrary integral domain without stable ideals.
@article{MZM_1974_16_3_a15,
author = {M. S. Burgin},
title = {Cancellation law and attainable classes of linear $\Omega$-algebras},
journal = {Matemati\v{c}eskie zametki},
pages = {467--478},
publisher = {mathdoc},
volume = {16},
number = {3},
year = {1974},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/MZM_1974_16_3_a15/}
}
M. S. Burgin. Cancellation law and attainable classes of linear $\Omega$-algebras. Matematičeskie zametki, Tome 16 (1974) no. 3, pp. 467-478. http://geodesic.mathdoc.fr/item/MZM_1974_16_3_a15/