# ring of sets

## 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

• $\varnothing\in\mathcal{F}$,

• 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 of certain lattices as rings or fields of sets are the following:

1. 1.
2. 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