Definitions. In any lattice, the ``sup'' of a subset is its least upper bound, if it exists. Thus a sup is the same thing as a possibly infinite join. The sup of the empty subset is 0. Correspondingly, the ``inf'' of a subset is its greatest lower bound, if it exists, and the inf of the empty subset is 1. A lattice is complete if every subset has a sup and inf.
It is easy to show that if every subset in a lattice has a sup, then the lattice is already complete.
Definition. In a lattice with 0, an atom is an element that covers 0. A lattice is atomic if every element is the sup of some set of atoms.