Cancellation law and attainable classes of linear $\Omega$-algebras

M. S. Burgin

Matematičeskie zametki, Tome 16 (1974) no. 3, pp. 467-478.

Résumé

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.