Abstract
We initiate the radical theory of algebras with B-action where B is a fixed Boolean ring. We consider lattices of classes of algebras defined in terms of ideals of B. In two special cases (universal classes of \(\omega \)-groups with B-action and idempotent algebras with B-action), these ideal-defined classes are sublattices of the lattice of radicals, and we characterise semisimplicity in such cases.
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