Extensive figure

In mathematics, extensiveness describes the property of a mapping to "enlarge" quantities. Correspondingly, intensive (also anti-extensive ) images “reduce” quantities.

definition

Be a partially ordered set . An illustration ${\ displaystyle (A, \ leq)}$

{\ displaystyle f \ colon \, A \ to A}

is called extensive if:

{\ displaystyle a \ leq f (a)}

for everyone .

{\ displaystyle a \ in A}

It is called intensive if:

{\ displaystyle f (a) \ leq a}

for everyone .

{\ displaystyle a \ in A}

Examples

The identity is extensive and intense, as always applies. ${\ displaystyle (A, \ leq)}$ ${\ displaystyle \ operatorname {id} _ {A} \ colon \, a \ mapsto a}$ ${\ displaystyle a \ leq a}$
By definition, hull operators are extensive and kernel operators are intensive on the power set of an arbitrary set with the set- theoretical inclusion as partial order.

Bourbaki-Kneser's fixed point theorem

After the fixed-point theorem of Bourbaki and Kneser any extensive mapping has already then a fixed point if strictly ordered inductively is. From this the lemma of Zorn can be proven with the help of the axiom of choice . ${\ displaystyle f \ colon A \ rightarrow A}$ ${\ displaystyle A}$

literature

Marcel Erné: Introduction to Order Theory . Bibliographisches Institut u. a., Mannheim u. a. 1982, ISBN 3-411-01638-8 .
Heinrich Werner: Introduction to general algebra (= BI university pocket books . Volume 120 ). Bibliographisches Institut, Mannheim u. a. 1978, ISBN 3-411-00120-8 .
Serge Lang : Algebra. 3rd edition, reprinted, with corrections. Addison-Wesley, Reading MA et al. a. 1993, ISBN 0-201-55540-9 .