Teori order: Perbedaan antara revisi

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An '''[[order-embedding]]''' is a function ''f'' between orders that is both order-preserving and order-reflecting. Examples for these definitions are found easily. For instance, the function that maps a natural number to its successor is clearly monotone with respect to the natural order. Any function from a discrete order, i.e. from a set ordered by the identity order "=", is also monotone. Mapping each natural number to the corresponding real number gives an example for an order embedding. The [[complement (set theory)|set complement]] on a [[powerset]] is an example of an antitone function.
 
An important question is when two orders are "essentially equal", i.e. when they are the same up to renaming of elements. '''[[Order isomorphism]]s''' are functions that define such a renaming. An order-isomorphism is a monotone [[bijective]] function that has a monotone inverse. This is equivalent to being a [[surjective]] order-embedding. Hence, the image ''f''(''P'') of an order-embedding is always isomorphic to ''P'', which justifies the termistilah "embedding".
 
A more elaborate type of functions is given by so-called '''[[Galois connection]]s'''. Monotone Galois connections can be viewed as a generalization of order-isomorphisms, since they constitute of a pair of two functions in converse directions, which are "not quite" inverse to each other, but that still have close relationships.
 
But category theory also has its impact on order theory on a larger scale. Classes of posets with appropriate functions as discussed above form interesting categories. Often one can also state constructions of orders, like the [[product order]], in terms of categories. Further insights result when categories of orders are found [[equivalence of categories|categorically equivalent]] to other categories, for example of topological spaces. This line of research leads to various ''representation theorems'', often collected under the label of [[Stone duality]].
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== Sejarah ==
AsSebagaimana explaineddijelaskan beforesebelumnya, orderstatanan aresangat ubiquitousbanyak inditemuai mathematicsdalam matematika. HoweverNamun, earliestpenyebutan expliciteksplisit mentioningspaling ofawal partialmengenai orderstatanan areparsial probablydapat todilacak besetelah foundabad not before the 19th centuryke-19. InDalam thiskonteks contextini the works ofkarya [[George Boole]] aredianggap ofsangat great importancepenting. Moreover,Di workssamping ofitu [[Charles Sanders Peirce]], [[Richard Dedekind]], anddan [[Ernst Schröder]] alsojuga considermembahas conceptskonsep ofteori order theory. Certainly, there are others to be named in this context and surely there exists more detailed material on the history of order theory. <!-- ''Please contribute if any further knowledge is available to you.'' -->
 
The termIstilah "''poset''" assebagai ansingkatan abbreviationdari for"''<u>p</u>artially partially ordered<u>o</u>rdered <u>set</u>''", wasyaitu coined"himpunan bydengan tatanan parsial", digagas oleh [[:en:Garrett Birkhoff|Garrett Birkhoff]] in the seconddalam editionedisi ofkedua hisbukunya influentialyang bookberpengaruh ''Lattice Theory''.<ref>Birkhoff 1948, p.1</ref><ref>[http://jeff560.tripod.com/o.html Earliest Known Uses of Some of the Words of Mathematics]</ref>
 
The term ''poset'' as an abbreviation for partially ordered set was coined by [[Garrett Birkhoff]] in the second edition of his influential book ''Lattice Theory''.<ref>Birkhoff 1948, p.1</ref><ref>[http://jeff560.tripod.com/o.html Earliest Known Uses of Some of the Words of Mathematics]</ref>
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== Lihat pula ==
* [[Cyclic order]]
* [[Hierarki]]
* [[Incidence algebra]]
* [[List of publications in mathematics#Order theory|Important publications in order theory]]
* [[Himpunan kausal]]