  We describe the basic lattice structures of attractors and repellers in
dynamical systems. The structure of distributive lattices allows for an
algebraic treatment of gradient-like dynamics in general dynamical systems,
both invertible and noninvertible. We separate those properties which rely
solely on algebraic structures from those that require some topological
arguments, in order to lay a foundation for the development of algorithms to
manipulate these structures computationally.
