Core operator

In mathematics , the kernel of a set is a subset that is small enough to meet certain requirements, and at the same time the largest set that meets these requirements. The most important example is the open core or the interior of a subset of a topological space . Core operator describes the rule by which every set of objects is assigned its core. The kernels given by a kernel operator form a kernel system , i.e. a set system with certain properties.

Definitions

Core operators

Over a given basic set , a kernel operator is an intensive , monotonic , idempotent mapping to the power set of , which assigns a further subset of , namely the kernel , to each subset , whereby the following conditions are met: ${\displaystyle X}$ ${\displaystyle K\colon \;{\mathcal {P}}(X)\to {\mathcal {P}}(X)}$ ${\displaystyle X}$ ${\displaystyle A\subseteq X}$ ${\displaystyle X}$ ${\displaystyle K(A)\subseteq X}$

(It) intensity: that is: the core of is contained at least in the set itself. ${\displaystyle K(A)\subseteq A}$ ${\displaystyle A}$ ${\displaystyle A}$

(M) Monotony or isotony:, that means: If is a subset of , this also applies accordingly to its nuclei. ${\displaystyle A\subseteq B\ \Rightarrow \ K(A)\subseteq K(B)}$ ${\displaystyle A}$ ${\displaystyle B}$

(Ip) Idempotence:, that means: If one forms the core of a set again, it remains unchanged. ${\displaystyle K(K(A))=K(A)}$

Because of the other two requirements, instead of idempotence, it is sufficient to just demand, that is: if you build the core of a set again, nothing is taken away. ${\displaystyle K(A)\subseteq K(K(A))}$

The following is equivalent to the three named individual claims. is called the core operator if the following applies to all : ${\displaystyle K\colon \;{\mathcal {P}}(X)\to {\mathcal {P}}(X)}$ ${\displaystyle A,B\subseteq X}$

(Ok) . ${\displaystyle K(A)\subseteq K(B)\Longleftrightarrow K(A)\subseteq B}$

Core systems

A kernel system is a set system closed under any union set formation, i. H. A core system over a set is a set consisting of subsets of the basic set with the following properties: ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {S}}}$

(Sk ₀ ): contains the empty set : . ${\displaystyle {\mathcal {S}}}$ ${\displaystyle \emptyset \in {\mathcal {S}}}$

(Sk ₁ ): For each non-empty subset of the union of the elements is an element of , or in short . ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle \forall \;{\mathcal {T}}\subseteq {\mathcal {S}},\,{\mathcal {T}}\neq \emptyset \colon \;\bigcup {\mathcal {T}}\in {\mathcal {S}}}$

Because of this, the two mentioned conditions can be simplified to a single equivalent condition: ${\displaystyle \bigcup \emptyset =\emptyset }$

(Sk): For each subset of the union of the elements is an element of , or in short . ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle \forall \;{\mathcal {T}}\subseteq {\mathcal {S}}\colon \;\bigcup {\mathcal {T}}\in {\mathcal {S}}}$

Relationship between core systems and core operators

Core systems and core operators correspond to one another:

• Is a core system , then you can a nuclear operator to define as follows: ${\displaystyle {\mathcal {S}}}$ ${\displaystyle X}$ ${\displaystyle K_{\mathcal {S}}}$ ${\displaystyle X}$

${\displaystyle K_{\mathcal {S}}(A):=\bigcup \{Y\in {\mathcal {S}}\mid Y\subseteq A\}}$for everyone . ${\displaystyle A\subseteq X}$

• Conversely, from any nuclear operator on a core system about to be won: ${\displaystyle K}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {S}}_{K}}$ ${\displaystyle X}$

${\displaystyle {\mathcal {S}}_{K}:=\{K(A)\mid A\subseteq X\}}$.

example

• The open sets of a topological space form a core system, namely the topology of the space. The associated core operator is the formation of the interior of a subset.

See also

• Hull operator

literature

• Marcel Erné: Introduction to Order Theory . Bibliographisches Institut, Mannheim 1982, ISBN 3-411-01638-8 .
• Heinrich Werner: Introduction to general algebra . Bibliographisches Institut, Mannheim 1978, ISBN 3-411-00120-8 .