ring of sets

ring of sets

Ring of Sets

Let

S

be a set and

2

S

be the

power set

of

S

. A subset

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,

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

•

ℱ

is a ring of sets of

S

,

•

S

∈

ℱ

, and

•

if

A

∈

ℱ

, then the

complement

A

¯

∈

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

A

¯

∈

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

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