Ring of sets

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Not to be confused with Ring (mathematics). ‹See Tfd›

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, for any A,B\in\mathcal{R},

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

In measure theory, a ring of sets is instead a family closed under unions and set-theoretic differences.[2] That is, it obeys the two properties

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

This implies that it is also closed under symmetric difference and intersection, because of the identities

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

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.

Examples[edit]

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.

The open sets and closed sets of any topological space are closed under both unions and intersections.[1]

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.[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[edit]

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

References[edit]

  1. ^ a b c d e 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[edit]