Ring of sets

Related structures

A ring of sets in the order-theoretic sense forms a distributive lattice in which the intersection and union operations correspond to the lattice's meet and join operations, respectively. Conversely, every distributive lattice is isomorphic to a ring of sets; in the case of finite distributive lattices, this is Birkhoff's representation theorem and the sets may be taken as the lower sets of a partially ordered set.

A family of sets closed under union and relative complement is also closed under symmetric difference and intersection. Conversely, every family of sets closed under both symmetric difference and intersection is also closed under union and relative complement. This is due to the identities

1. ${\displaystyle A\cup B=(A\mathrm{△}B)\mathrm{△}(A\cap B)}$ and
2. ${\displaystyle A\setminus B=A\mathrm{△}(A\cap B).}$

Symmetric difference and intersection together give a ring in the measure-theoretic sense the structure of a boolean ring.

In the measure-theoretic sense, a σ-ring is a ring closed under countable unions, and a δ-ring is a ring closed under countable intersections. Explicitly, a σ-ring over ${\displaystyle X}$ is a set ${\displaystyle \mathcal{F}}$ such that for any sequence ${\displaystyle \{A_k{\}}_{k=1}^{\mathrm{\infty }}\subseteq \mathcal{F},}$ we have $\bigcup _{k=1}^{\mathrm{\infty }}{A}_k\in \mathcal{F}.$

Given a set ${\displaystyle X,}$ a field of sets − also called an algebra over ${\displaystyle X}$ − is a ring that contains ${\displaystyle X.}$ This definition entails that an algebra is closed under absolute complement ${\displaystyle A^c=X\setminus A.}$ A σ-algebra is an algebra that is also closed under countable unions, or equivalently a σ-ring that contains ${\displaystyle X.}$ In fact, by de Morgan's laws, a δ-ring that contains ${\displaystyle X}$ is necessarily a σ-algebra as well. Fields of sets, and especially σ-algebras, are central to the modern theory of probability and the definition of measures.

A semiring (of sets) is a family of sets ${\displaystyle \mathcal{S}}$ with the properties

1. ${\displaystyle \varnothing \in \mathcal{S},}$
   • If (3) holds, then ${\displaystyle \varnothing \in \mathcal{S}}$ if and only if ${\displaystyle \mathcal{S}\ne \varnothing .}$
2. ${\displaystyle A,B\in \mathcal{S}}$ implies ${\displaystyle A\cap B\in \mathcal{S},}$ and
3. ${\displaystyle A,B\in \mathcal{S}}$ implies ${\displaystyle A\setminus B=\bigcup _{i=1}^n{C}_i}$ for some disjoint ${\displaystyle C_1,\dots ,C_n\in \mathcal{S}.}$

Every ring (in the measure theory sense) is a semi-ring. On the other hand, ${\displaystyle \mathcal{S}:=\{\mathrm{\varnothing },\{x\},\{y\}\}}$ on ${\displaystyle X=\{x,y\}}$ is a semi-ring but not a ring, since it is not closed under unions.

A semialgebra or elementary family is a collection ${\displaystyle \mathcal{S}}$ of subsets of ${\displaystyle X}$ satisfying the semiring properties except with (3) replaced with:

• If ${\displaystyle E\in \mathcal{S}}$ then there exists a finite number of mutually disjoint sets ${\displaystyle C_1,\dots ,C_n\in \mathcal{S}}$ such that ${\displaystyle X\setminus E=\bigcup _{i=1}^n{C}_i.}$

This condition is stronger than (3), which can be seen as follows. If ${\displaystyle \mathcal{S}}$ is a semialgebra and ${\displaystyle E,F\in \mathcal{S}}$, then we can write ${\displaystyle F^c=F_1\cup \dots \cup F_n}$ for disjoint ${\displaystyle F_i\in S}$. Then: $${\displaystyle E\setminus F=E\cap F^c=E\cap (F_1\cup \dots \cup F_n)=(E\cap F_1)\cup \dots \cup (E\cap F_n)}$$

and every ${\displaystyle E\cap F_i\in S}$ since it is closed under intersection, and disjoint since they are contained in the disjoint ${\displaystyle F_i}$'s. Moreover the condition is strictly stronger: any ${\displaystyle S}$ that is both a ring and a semialgebra is an algebra, hence any ring that is not an algebra is also not a semialgebra (e.g. the collection of finite sets on an infinite set ${\displaystyle X}$).