On distributivity of the lattice of subquasivarieties of a locally finite semi-simple arithmetical variety (Q799716)

Theorem: The lattice of all subvarieties of a locally finite semi-simple arithmetical variety \({\mathbb{K}}\) is distributive iff \(s({\mathbb{K}})\) is a chain and an initial segment of \(S({\mathbb{K}})\), where \(S({\mathbb{K}})\) denotes the set of isomorphism types of finite simple algebras from \({\mathbb{K}}\) and \(s({\mathbb{K}})\) denotes the set of types of algebras from \(S({\mathbb{K}})\), which contain no trivial algebra.