ring of sets
Ring of Sets
Let S be a set and 2S be the power set of S. A subset ℛ of 2S is said to be a ring of sets of S if it is a lattice
under the intersection
and union operations
. In other words, ℛ is a ring of sets if
-
•
for any A,B∈ℛ, then A∩B∈ℛ,
-
•
for any A,B∈ℛ, then A∪B∈ℛ.
A ring of sets is a distributive lattice. The word “ring” in the name has nothing to do with the ordinary ring found in algebra
. Rather, it is an abelian semigroup with respect to each of the binary set operations. If S∈ℛ, then (ℛ,∩,S) becomes an abelian
monoid. Similarly, if ∅∈ℛ, then (ℛ,∪,∅) is an abelian monoid. If both S,∅∈ℛ, then (ℛ,∪,∩) is a commutative semiring
, since ∅∩A=A∩∅=∅, and ∩ distributes over ∪. Dualizing, we see that (ℛ,∩,∪) 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 ℛ, a ring of sets can in theory be the empty set. Even if ℛ 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∈ℛ, then A-B∈ℛ.
This is indeed a stronger condition, as A∩B=A-(A-B)∈ℛ. However, we shall stick with the more general definition here.
Field of Sets
An even stronger condition is to insist that not only is ℛ non-empty, but that S∈ℛ. Such a ring of sets is called a field, or algebra of sets. Formally, given a set S, a field of sets ℱ of S satisfies the following criteria
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•
ℱ is a ring of sets of S,
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•
S∈ℱ, and
-
•
if A∈ℱ, then the complement
ˉA∈ℱ.
The three conditions above are equivalent to the following three conditions:
-
•
∅∈ℱ,
-
•
if A,B∈ℱ, then A∪B∈ℱ, and
-
•
if A∈ℱ, then ˉA∈ℱ.
A field of sets is also known as an algebra of sets.
It is easy to see that ℱ is a distributive complemented lattice, and hence a Boolean lattice. From the discussion earlier, we also see that ℱ (of S) is a commutative semiring, with S acting as the multiplicative identity and ∅ both the additive identity and the multiplicative absorbing element.
Remark. Two remarkable theorems relating to of certain lattices as rings or fields of sets are the following:
-
1.
a lattice is distributive iff it is lattice isomorphic
(http://planetmath.org/LatticeIsomorphism) to a ring of sets (G. Birkhoff and M. Stone);
-
2.
a lattice is Boolean (http://planetmath.org/BooleanLattice) iff it is lattice to a field of sets (M. Stone).
References
- 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 |