Ring of sets
In mathematics, there are two different notions of a ring of sets, both referring to certain families of sets. In order theory, a nonempty family of sets is called a ring (of sets) if it is closed under intersection and union. That is, for any ,
This implies that it is also closed under symmetric difference and intersection, because of the identities
Together, these operations give the structure of a boolean ring. Conversely, every family of sets closed under both symmetric difference and intersection is also closed under union and differences.
If X is any set, then the power set of X (the family of all subsets of X) forms a ring of sets in either sense.
If (X,≤) is a partially ordered set, then its upper sets (the subsets of X with the additional property that if x belongs to an upper set U and x ≤ y, then y must also belong to U) is closed under both intersections and unions. However, in general it will not be closed under differences of sets.
If T is any transformation of a space, then the sets that are mapped into themselves by T are closed under both unions and intersections.
If two rings of sets are both defined on the same elements, then the sets that belong to both rings themselves form a ring of sets.
A ring of sets (in the order-theoretic sense) 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.