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.