Page "Ring of sets" Paragraph 8

A ring of sets forms a distributive lattice in which the intersection and union operations correspond to the lattice's meet and join operations, respectively.

Conversely, every distributive lattice is isomorphic to a ring of sets ; in the case of finite distributive lattices, this is Birkhoff's representation theorem and the sets may be taken as the lower sets of a partially ordered set.

Every field of sets and so also any σ-algebra also is a ring of sets.