Ring (set theory)

In set theory, a ring is a nonempty set system R that is closed with respect to the intersection and the symmetric difference of a finite number of elements. This means that for any elements A , B from the ring, the elements $A\cap B$ ${\ displaystyle A \ cap B}$ $A \ cap B$ and $A\triangle B$ ${\ displaystyle A \ triangle B}$ $A \ triangle B$ will also lie in the ring.

Content

1 Ring of sets as an algebraic ring
2 Properties of rings
3 Extensions and restrictions of the concept
4 Examples
5 See also
6 notes

Ring of sets as an algebraic ring

From the point of view of the algebraic structure, the ring of sets is an associative commutative ring with the operation of a symmetric difference in the role of addition and intersection in the role of multiplication. The neutral element in addition is obviously an empty set . There may not be a neutral element for multiplication in the ring of sets. For example, the ring of all bounded subsets of the number line does not have a neutral element for multiplication ^[1] .

Ring Properties

An empty set belongs to any ring (since $\varnothing =A\triangle A$ ${\ displaystyle \ varnothing = A \ triangle A}$ )
The union of a finite number of elements of the ring belongs to the ring, since $A\cup B=(A\triangle B)\triangle (A\cap B)$ ${\ displaystyle A \ cup B = (A \ triangle B) \ triangle (A \ cap B)}$ .
The difference of the elements of the ring also belongs to the ring, since $A\backslash B=A\triangle (A\cap B)$ ${\ displaystyle A \ backslash B = A \ triangle (A \ cap B)}$ .

Extensions and Narrowings of the Concept

A ring is a special case of a semiring. Moreover, each half-ring can be turned into a ring by adding a certain number of elements. A minimal ring generated by a given semiring S is an R such that any ring containing S contains it. For each semiring S such R exists and is unique; it consists of all possible finite unions of elements of S.

An algebra is a ring with unity, that is, an element E such that the intersection of E with any element A is equal to A. A sigma ring is a ring closed with respect to countable unions of elements, and a delta ring is a ring closed with respect to countable intersections. A sigma algebra is defined similarly (in addition, any delta algebra is a sigma algebra and vice versa).

Examples

Borel sigma-algebra of sets on the line
Bulean

Notes

↑ Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis. M .: Fizmatlit, 2009 - p. 48

[1] Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis. M .: Fizmatlit, 2009 - p. 48