PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
ring of sets (Definition)

Ring of Sets

Let $ S$ be a set and $ 2^S$ be the power set of $ S$. A subset $ \mathcal{R}$ of $ 2^S$ is said to be a ring of sets of $ S$ if it is a lattice under the intersection and union operations. In other words, $ \mathcal{R}$ is a ring of sets if
  • for any $ A,B\in \mathcal{R}$, then $ A\cap B\in \mathcal{R}$,
  • for any $ A,B\in \mathcal{R}$, then $ A\cup B\in \mathcal{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\in\mathcal{R}$, then $ (\mathcal{R},\cap,S)$ becomes an abelian monoid. Similarly, if $ \varnothing\in\mathcal{R}$, then $ (\mathcal{R},\cup,\varnothing)$ is an abelian monoid. If both $ S,\varnothing\in\mathcal{R}$, then $ (\mathcal{R},\cup,\cap)$ is a commutative semiring, since $ \varnothing\cap A=A\cap\varnothing=\varnothing$, and $ \cap$ distributes over $ \cup$. Dualizing, we see that $ (\mathcal{R},\cap,\cup)$ 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 $ \mathcal{R}$, a ring of sets can in theory be the empty set. Even if $ \mathcal{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\in \mathcal{R}$, then $ A-B\in \mathcal{R}$.
This is indeed a stronger condition, as $ A\cap B=A-(A-B)\in \mathcal{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 $ \mathcal{R}$ non-empty, but that $ S\in\mathcal{R}$. Such a ring of sets is called a field, or algebra of sets. Formally, given a set $ S$, a field of sets $ \mathcal{F}$ of $ S$ satisfies the following criteria
  • $ \mathcal{F}$ is a ring of sets of $ S$,
  • $ S\in\mathcal{F}$, and
  • if $ A\in\mathcal{F}$, then the complement $ \overline{A}\in\mathcal{F}$.
The three conditions above are equivalent to the following three conditions:
  • $ \varnothing\in\mathcal{F}$,
  • if $ A,B\in \mathcal{F}$, then $ A\cup B\in \mathcal{F}$, and
  • if $ A\in\mathcal{F}$, then $ \overline{A}\in\mathcal{F}$.

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

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

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

  1. a lattice is distributive iff it is lattice isomorphic to a ring of sets (G. Birkhoff and M. Stone);
  2. a lattice is Boolean iff it is lattice isomorphic to a field of sets (M. Stone).

Bibliography

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).



"ring of sets" is owned by CWoo. [ full author list (2) ]
(view preamble)

View style:

See Also: $\sigma$-algebra, absorbing element

Other names:  lattice of sets, algebra of sets
Also defines:  field of sets

Attachments:
finite fields of sets (Theorem) by rspuzio
examples of ring of sets (Example) by rspuzio
Log in to rate this entry.
(view current ratings)

Cross-references: iff, absorbing element, multiplicative, identity, additive, multiplicative identity, Boolean lattice, complemented lattice, distributive, easy to see, equivalent, complement, satisfies, field, measure, singleton, even, empty set, theory, connection, distributes over, semiring, commutative, monoid, abelian, binary, Abelian semigroup, algebra, ring, distributive lattice, operations, union, intersection, lattice, subset, power set
There are 8 references to this entry.

This is version 15 of ring of sets, born on 2006-03-21, modified 2007-12-18.
Object id is 7757, canonical name is RingOfSets.
Accessed 3293 times total.

Classification:
AMS MSC: 28A05 (Measure and integration :: Classical measure theory :: Classes of sets , measurable sets, Suslin sets, analytic sets)
 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
Pending Errata and Addenda
None.
[ View all 3 ]
Discussion
Style: Expand: Order:
forum policy
Definition by Andrea Ambrosio on 2006-05-16 16:07:17
Reading various books and lectures on Measure Theory, I have never been able to find your definition of ring of sets.

Namely, I find always two definitions,

a) A ring is a (non-empty) collection of sets which is closed under finite difference (A\B) and finite union.

b) A ring is a (non-empty) collection of sets which is closed under finite symmetric difference (A /\ B) and finite intersection.

In both cases it's easy to prove that a ring is closed also for finite intersection (def a) and for finite union (def b).

On the other side, in order to prove the equivalence of the definition, I tried to use your definition to prove that a ring is closed under any type of difference (ordinary and symmetric), but I wasn't able to, and sincerely I think it's not possible.

Am I missing something? Could you clarify please?

Thank you and best regards,

Andrea
[ reply | up ]
Interact
post | correct | update request | add derivation | add example | add (any)