  We study a two-parameter generalization of the Freese-Nation Property of
boolean algebras and its order-theoretic and topological consequences.
  For every regular infinite \kappa, the (\kappa,\kappa)-FN, the
(\kappa^+,\kappa)-FN, and the \kappa-FN are known to be equivalent; we show
that the family of properties (\lambda,\mu)-FN for \lambda>\mu form a true
two-dimensional hierarchy that is robust with respect to coproducts, retracts,
and the exponential operation.
  The (\kappa,\aleph_0)-FN in particular has strong consequences for base
properties of compacta (stronger still for homogeneous compacta), and these
consequences have natural duals in terms of special subsets of boolean
algebras. We show that the (\kappa,\aleph_0)-FN also leads to a generalization
of the equality of weight and \pi-character in dyadic compacta.
  Elementary subalgebras and their duals, elementary quotient spaces, were
originally used to define the (\lambda, \kappa)-FN and its topological dual,
which naturally generalized from Stone spaces to all compacta, thereby
generalizing Shchepin's notion of openly generated compacta. We introduce a
simple combinatorial definition of the (\lambda, \kappa)-FN that is equivalent
to the original for regular infinite cardinals \lambda>\kappa.
