A variety V of Boolean algebras with operators is singleton-persistent if it contains a complex algebra whenever it contains the subalgebra generated by the singletons. V is atom-canonical if it contains the complex algebra of the atom structure of any of the atomic members of V.This paper explores relationships between these "persistence" properties and questions of whether V is generated by its complex algebras or its atomic members, or is closed under canonical embedding algebras or completions. It also develops a (...) general theory of when operations involving complex algebras lead to the construction of elementary classes of relational structures. (shrink)