ring of sets

Ring of Sets

Let S be a set and 2S be the power setMathworldPlanetmath of S. A subset of 2S is said to be a ring of sets of S if it is a latticeMathworldPlanetmath under the intersectionMathworldPlanetmath and union operationsMathworldPlanetmath. In other words, is a ring of sets if

  • for any A,B, then AB,

  • for any A,B, then AB.

A ring of sets is a distributive latticeMathworldPlanetmath. The word “ring” in the name has nothing to do with the ordinary ring found in algebraPlanetmathPlanetmathPlanetmath. Rather, it is an abelian semigroup with respect to each of the binary set operations. If S, then (,,S) becomes an abelianMathworldPlanetmathPlanetmath monoid. Similarly, if , then (,,) is an abelian monoid. If both S,, then (,,) is a commutative semiringMathworldPlanetmath, since A=A=, and distributes over . Dualizing, we see that (,,) is also a commutative semiring. It is perhaps with this connection that the name “ring of sets” is so chosen.

Since S is not required to be in , a ring of sets can in theory be the empty setMathworldPlanetmath. Even if may be non-empty, it may be a singleton. Both cases are not very interesting to study. To avoid such examples, some authors, particularly measure theorists, define a ring of sets to be a non-empty set with the first condition above replaced by

  • for any A,B, then A-B.

This is indeed a stronger condition, as AB=A-(A-B). However, we shall stick with the more general definition here.

Field of Sets

An even stronger condition is to insist that not only is non-empty, but that S. Such a ring of sets is called a field, or algebra of sets. Formally, given a set S, a field of sets of S satisfies the following criteria

  • is a ring of sets of S,

  • S, and

  • if A, then the complementPlanetmathPlanetmath A¯.

The three conditions above are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the following three conditions:

  • ,

  • if A,B, then AB, and

  • if A, then A¯.

A field of sets is also known as an algebra of sets.

It is easy to see that is a distributive complemented lattice, and hence a Boolean lattice. From the discussion earlier, we also see that (of S) is a commutative semiring, with S acting as the multiplicative identityPlanetmathPlanetmath and both the additive identity and the multiplicative absorbing element.

Remark. Two remarkable theoremsMathworldPlanetmath relating to of certain lattices as rings or fields of sets are the following:

  1. 1.

    a lattice is distributive iff it is lattice isomorphicPlanetmathPlanetmathPlanetmath (http://planetmath.org/LatticeIsomorphism) to a ring of sets (G. Birkhoff and M. Stone);

  2. 2.

    a lattice is Boolean (http://planetmath.org/BooleanLattice) iff it is lattice to a field of sets (M. Stone).


  • 1 P. R. Halmos: Lectures on Boolean Algebras, Springer-Verlag (1970).
  • 2 P. R. Halmos: Measure Theory, Springer-Verlag (1974).
  • 3 G. Grätzer: General Lattice Theory, Birkhäuser, (1998).
Title ring of sets
Canonical name RingOfSets
Date of creation 2013-03-22 15:47:46
Last modified on 2013-03-22 15:47:46
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 18
Author CWoo (3771)
Entry type Definition
Classification msc 03E20
Classification msc 28A05
Synonym lattice of sets
Synonym algebra of sets
Related topic SigmaAlgebra
Related topic AbsorbingElement
Related topic RepresentingADistributiveLatticeByRingOfSets
Related topic RepresentingABooleanLatticeByFieldOfSets
Defines field of sets