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 $\mathcal{R}$ is called a ring (of sets) if it is closed under intersection and union. That is, the following two statements are true for all sets $A$ and $B$,

1. $A,B\in\mathcal{R}$ implies $A \cap B \in \mathcal{R}$ and
2. $A,B\in\mathcal{R}$ implies $A \cup B \in \mathcal{R}.$^[1]

In measure theory, a ring of sets $\mathcal{R}$ is instead a nonempty family closed under unions and set-theoretic differences.^[2] That is, the following two statements are true for all sets $A$ and $B$ (including when they are the same set),

1. $A,B\in\mathcal{R}$ implies $A \setminus B \in \mathcal{R}$ and
2. $A,B\in\mathcal{R}$ implies $A \cup B \in \mathcal{R}.$

This implies the empty set is in $\mathcal{R}$. It also implies that $\mathcal{R}$ is closed under symmetric difference and intersection, because of the identities

1. $A\,\triangle\,B = (A \setminus B) \cup (B \setminus A)$ and
2. $A\cap B=A\setminus(A\setminus B).$

(So a ring in the second, measure theory, sense is also a ring in the first, order theory, sense.) Together, these operations give $\mathcal{R}$ 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. This is due to the identities

1. $A \cup B = (A\, \triangle\, B)\, \triangle\, (A \cap B)$ and
2. $A \setminus B = A\, \triangle\, (A \cap B).$

Examples

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) 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 intervals of the form (a, b], a,b in R is a ring in the measure theory 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]

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.^[1]

Related structures

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 field of subsets of X is a ring that contains X and is closed under relative complement. Every field, and so also every σ-algebra, is a ring of sets in the measure theory sense.

A semi-ring (of sets) is a family of sets $\mathcal{S}$ with the properties

1. $\emptyset \in \mathcal{S},$
2. $A, B \in \mathcal{S}$ implies $A \cap B \in \mathcal{S},$ and
3. $A, B \in \mathcal{S}$ implies $A \setminus B = \bigcup_{i=1}^nC_i$ for some disjoint $C_1,\dots,C_n \in \mathcal{S}.$

Clearly, every ring (in the measure theory sense) is a semi-ring.

A semi-field of subsets of X is a semi-ring that contains X.

References

1. Birkhoff, Garrett (1937), "Rings of sets", Duke Mathematical Journal 3 (3): 443–454, doi:10.1215/S0012-7094-37-00334-X, MR 1546000.
2. De Barra, Gar (2003), Measure Theory and Integration, Horwood Publishing, p. 13, ISBN 9781904275046.

External links

• Ring of sets at Encyclopedia of Mathematics