In order theory, a nonempty family of sets
is called a ring (of sets) if it is closed under union and intersection.
, In measure theory, a nonempty family of sets
is called a ring (of sets) if it is closed under union and relative complement (set-theoretic difference).
, This implies that a ring in the measure-theoretic sense always contains the empty set.
Furthermore, for all sets A and B, which shows that a family of sets closed under relative complement is also closed under intersection, so that a ring in the measure-theoretic sense is also a ring in the order-theoretic 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) are closed under both intersections and unions.
However, in general it will not be closed under differences of sets.
The open sets and closed sets of any topological space are closed under both unions and intersections.
[1] On the real line R, the family of sets consisting of the empty set and all finite unions of half-open intervals of the form (a, b], with a, b ∈ R is a ring in the measure-theoretic sense.
If T is any transformation defined on a space, then the sets that are mapped into themselves by T are closed under both unions and intersections.
[1] 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.
[1] A family of sets closed under union and relative complement is also closed under symmetric difference and intersection.
Conversely, every family of sets closed under both symmetric difference and intersection is also closed under union and relative complement.
This is due to the identities Symmetric difference and intersection together give a ring in the measure-theoretic sense the structure of a boolean ring.
In the measure-theoretic sense, a σ-ring is a ring closed under countable unions, and a δ-ring is a ring closed under countable intersections.
a field of sets − also called an algebra over
This definition entails that an algebra is closed under absolute complement
A σ-algebra is an algebra that is also closed under countable unions, or equivalently a σ-ring that contains
In fact, by de Morgan's laws, a δ-ring that contains
Fields of sets, and especially σ-algebras, are central to the modern theory of probability and the definition of measures.
with the properties Every ring (in the measure theory sense) is a semi-ring.
is a semi-ring but not a ring, since it is not closed under unions.
A semialgebra[3] or elementary family [4] is a collection
satisfying the semiring properties except with (3) replaced with: This condition is stronger than (3), which can be seen as follows.
that is both a ring and a semialgebra is an algebra, hence any ring that is not an algebra is also not a semialgebra (e.g. the collection of finite sets on an infinite set
Additionally, a semiring is a π-system where every complement
is equal to a finite disjoint union of sets in
is equal to a finite disjoint union of sets in