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
is called extensive if:
- for everyone .
It is called intensive if:
- for everyone .
Examples
- The identity is extensive and intense, as always applies.
- 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 .
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 .