Delta-ring

For p {\displaystyle p} -derivations used in commutative algebra to define prismatic cohomology, see P-derivation.

In mathematics, a non-empty collection of sets R {\displaystyle {\mathcal {R}}} 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, "Durchschnitt", 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 R {\displaystyle {\mathcal {R}}} is called a δ-ring if it has all of the following properties:

1. Closed under finite unions: A ∪ B ∈ R {\displaystyle A\cup B\in {\mathcal {R}}} for all A , B ∈ R , {\displaystyle A,B\in {\mathcal {R}},}
2. Closed under relative complementation: A − B ∈ R {\displaystyle A-B\in {\mathcal {R}}} for all A , B ∈ R , {\displaystyle A,B\in {\mathcal {R}},} and
3. Closed under countable intersections: ⋂ n = 1 ∞ A n ∈ R {\displaystyle \bigcap _{n=1}^{\infty }A_{n}\in {\mathcal {R}}} if A n ∈ R {\displaystyle A_{n}\in {\mathcal {R}}} for all n ∈ N . {\displaystyle n\in \mathbb {N} .}

If only the first two properties are satisfied, then R {\displaystyle {\mathcal {R}}} 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 K = { S ⊆ R : S  is bounded } {\displaystyle {\mathcal {K}}=\{S\subseteq \mathbb {R} :S{\text{ is bounded}}\}} is a δ-ring but not a 𝜎-ring because ⋃ n = 1 ∞ [ 0 , n ] {\textstyle \bigcup _{n=1}^{\infty }[0,n]} 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 – Measure theory and probability theoremPages displaying short descriptions of redirect targets
• 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 F {\displaystyle {\mathcal {F}}} of sets over Ω {\displaystyle \Omega } v t e
Is necessarily true of F : {\displaystyle {\mathcal {F}}\colon } or, is F {\displaystyle {\mathcal {F}}} closed under:	Directed by ⊇ {\displaystyle \,\supseteq }	A ∩ B {\displaystyle A\cap B}	A ∪ B {\displaystyle A\cup B}	B ∖ A {\displaystyle B\setminus A}	Ω ∖ A {\displaystyle \Omega \setminus A}	A 1 ∩ A 2 ∩ ⋯ {\displaystyle A_{1}\cap A_{2}\cap \cdots }	A 1 ∪ A 2 ∪ ⋯ {\displaystyle A_{1}\cup A_{2}\cup \cdots }	Ω ∈ F {\displaystyle \Omega \in {\mathcal {F}}}	∅ ∈ F {\displaystyle \varnothing \in {\mathcal {F}}}	F.I.P.
π-system
Semiring										Never
Semialgebra (semifield)										Never
Monotone class						only if A i ↘ {\displaystyle A_{i}\searrow }	only if A i ↗ {\displaystyle A_{i}\nearrow }
𝜆-system (Dynkin system)				only if A ⊆ B {\displaystyle A\subseteq B}			only if A i ↗ {\displaystyle A_{i}\nearrow } 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
Filter
Proper filter				Never	Never				Never
Prefilter (filter base)
Filter subbase
Open topology							(even arbitrary ∪ {\displaystyle \cup } )			Never
Closed topology						(even arbitrary ∩ {\displaystyle \cap } )				Never
Is necessarily true of F : {\displaystyle {\mathcal {F}}\colon } or, is F {\displaystyle {\mathcal {F}}} closed under:	directed downward	finite intersections	finite unions	relative complements	complements in Ω {\displaystyle \Omega }	countable intersections	countable unions	contains Ω {\displaystyle \Omega }	contains ∅ {\displaystyle \varnothing }	Finite intersection property
Additionally, a semiring is a π-system where every complement B ∖ A {\displaystyle B\setminus A} is equal to a finite disjoint union of sets in F . {\displaystyle {\mathcal {F}}.} A semialgebra is a semiring where every complement Ω ∖ A {\displaystyle \Omega \setminus A} is equal to a finite disjoint union of sets in F . {\displaystyle {\mathcal {F}}.} A , B , A 1 , A 2 , … {\displaystyle A,B,A_{1},A_{2},\ldots } are arbitrary elements of F {\displaystyle {\mathcal {F}}} and it is assumed that F ≠ ∅ . {\displaystyle {\mathcal {F}}\neq \varnothing .}

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