Mass association

In mathematics one is set lattice , a fundamental concept of measure theory and lattice theory . It denotes a non-empty system of sets that is unified and stable on average .

Felix Hausdorff called a set “ring” because of “an approximate analogy” to the algebraic structure of a ring in algebraic number theory . In measure theory today, however, a ring is understood to be a special set, because it is closely related to a ring in the sense of algebra - in contrast to a general set.

definition

Be any set. A system of subsets of is a lot of association or through association if the following criteria are fulfilled: ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$

1. ${\displaystyle {\mathcal {V}}\neq \emptyset }$( is not empty). ${\displaystyle {\mathcal {V}}}$
2. ${\displaystyle A,B\in {\mathcal {V}}\Rightarrow A\cup B\in {\mathcal {V}}}$(Stability / isolation with regard to union ).
3. ${\displaystyle A,B\in {\mathcal {V}}\Rightarrow A\cap B\in {\mathcal {V}}}$( Stability / seclusion with respect to average ).

Examples

• About any amount is a very small and with the power set given the greatest possible amount Association. ${\displaystyle \Omega }$ ${\displaystyle \{A\},A\subseteq \Omega ,}$ ${\displaystyle {\mathcal {P}}(\Omega )}$
• Every σ-algebra is a set (but not every set is a σ-algebra).

properties

• From the union and average stability it follows inductively that every non-empty, finite union and every non-empty, finite intersection of elements of the set is contained in it, i.e. H. applies to all : ${\displaystyle {\mathcal {V}}}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle A_{1},\dots ,A_{n}\in {\mathcal {V}}\Rightarrow A_{1}\cup \dots \cup A_{n}\in {\mathcal {V}}}$ and ${\displaystyle A_{1}\cap \dots \cap A_{n}\in {\mathcal {V}}.}$

Equivalent Definitions

If is a system of subsets of , then the following statements are equivalent : ${\displaystyle {\mathcal {V}}}$ ${\displaystyle \Omega }$

• ${\displaystyle {\mathcal {V}}}$ is a group.
• ${\displaystyle ({\mathcal {V}},\cup )}$and are semi-lattices in the sense of algebra . ${\displaystyle ({\mathcal {V}},\cap )}$
• ${\displaystyle ({\mathcal {V}},\cup ,\cap )}$is an association in the sense of algebra .
• ${\displaystyle ({\mathcal {V}},\cup ,\cap )}$is a distributive association in the sense of algebra .
• ${\displaystyle ({\mathcal {V}},\cup ,\cap )}$is an idempotent commutative half-ring in the sense of algebra .
• ${\displaystyle ({\mathcal {V}},\cup ,\cap )}$ is a half ring in the sense of algebra.

Related structures

• A set ring is a set that is also differential stable .
• A set algebra is a set that is even complement-stable . Set algebras are special set rings.

See also

• Set theory

literature

• Marcel Erné: Introduction to Order Theory . Bibliographisches Institut , Mannheim 1982, ISBN 3-411-01638-8 .
• U. Hebisch, H. J. Weinert: Half Rings - Algebraic Theory and Applications in Computer Science . Teubner , Stuttgart 1993, ISBN 3-519-02091-2 .
• Ernst Henze : Introduction to Dimension Theory . 2. revised Edition. Bibliographisches Institut, Mannheim / Zurich 1985, ISBN 3-411-03102-6 .
• Hans Hermes : Introduction to Association Theory . 2nd ext. Edition. Springer , Berlin / Heidelberg 1967.

Notes and individual references

1. ↑ Felix Hausdorff: Fundamentals of set theory . Veit & Comp., Leipzig 1914, p. 14 . Hausdorff referred to the association as "sum".
2. ↑ The term half-ring used here differs fundamentally from that of a (set) half-ring in the sense of measure theory , i.e. a special set system, the two are not related!