For -derivations used in commutative algebra to define prismatic cohomology, see P-derivation.

In mathematics, a non-empty collection of sets is called a δ-ring (pronounced "delta-ring") if it is closed under union, relative complementation, and countable intersection. The name "delta-ring" originates from the German word for intersection, "Durschnitt", which is meant to highlight the ring's closure under countable intersection, in contrast to a 𝜎-ring which is closed under countable unions.

A family of sets is called a δ-ring if it has all of the following properties:

1. Closed under finite unions: for all
2. Closed under relative complementation: for all and
3. Closed under countable intersections: if for all

If only the first two properties are satisfied, then is a ring of sets but not a δ-ring. Every 𝜎-ring is a δ-ring, but not every δ-ring is a 𝜎-ring.

δ-rings can be used instead of σ-algebras in the development of measure theory if one does not wish to allow sets of infinite measure.

The family is a δ-ring but not a 𝜎-ring because is not bounded.

• Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
• 𝜆-system (Dynkin system) – Family closed under complements and countable disjoint unions
• Monotone class – theoremPages displaying wikidata descriptions as a fallbackPages displaying short descriptions with no spaces
• π-system – Family of sets closed under intersection
• Ring of sets – Family closed under unions and relative complements
• σ-algebra – Algebraic structure of set algebra
• 𝜎-ideal – Family closed under subsets and countable unions
• 𝜎-ring – Family of sets closed under countable unions

• Cortzen, Allan. "Delta-Ring." From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/Delta-Ring.html

Families of sets over
Is necessarily true of or, is closed under:	Directed by									F.I.P.
π-system
Semiring										Never
Semialgebra (Semifield)										Never
Monotone class						only if	only if
𝜆-system (Dynkin System)				only if			only if or they are disjoint			Never
Ring (Order theory)
Ring (Measure theory)										Never
δ-Ring										Never
𝜎-Ring										Never
Algebra (Field)										Never
𝜎-Algebra (𝜎-Field)										Never
Dual ideal
Filter				Never	Never
Prefilter (Filter base)				Never	Never
Filter subbase				Never	Never
Open Topology							(even arbitrary )			Never
Closed Topology						(even arbitrary )				Never
Is necessarily true of or, is closed under:	directed downward	finite intersections	finite unions	relative complements	complements in	countable intersections	countable unions	contains	contains	Finite Intersection Property
Additionally, a semiring is a π-system where every complement is equal to a finite disjoint union of sets in A semialgebra is a semiring where every complement is equal to a finite disjoint union of sets in are arbitrary elements of and it is assumed that