Lattice ideal

Summary

An ideal of a lattice is a nonempty subset that is closed downward and closed under joins. The typical example is the principal ideal $\downarrow a = \set{x : x \le a}$ of all elements below a fixed element $a$. Lattice ideals are the order-theoretic analogue of ideals of a ring, and the dual notion is the filter. Prime ideals of lattices are the key to representation theorems for distributive lattices and Boolean algebras.