Radical classes of algebras with \(B\)-action (Q1966152)

The author develops the radical theory of algebras with \(B\)-action where \(B\) is a fixed Boolean ring. Algebras with \(B\)-actions were introduced by \textit{G. M. Bergman} [Algebra Univers. 28, No. 2, 153-187 (1991; Zbl 0718.18004)]. Such a notion has many natural interesting examples: -- any algebra \(\mathcal A\) is an algebra with \(B\)-action for \(B = \{0,1\}\), -- any cartesian product of algebras of the same signature, or -- bounded Boolean powers can be regarded as algebras with \(B\)-action. Lattices of classes of algebras defined in terms of ideals of \(B\) are considered. It is shown that in two special cases of universal classes (\(\Omega \)-groups with \(B\)-action and idempotent algebras with \(B\)-action) these ideal-defined classes are sublattices of the lattice of radicals, and semisimplicity in these cases is characterized.