Teori order: Perbedaan antara revisi
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<!--{{Outline|Outline of order theory}}-->
'''Teori order''' ({{lang-en|order theory}}) atau '''teori tatanan''' dan '''teori urutan''' (= teori keteraturan) adalah suatu cabang [[matematika]] yang meneliti pandangan intuitif manusia terhadap tatanan atau keteraturan dengan menggunakan hubungan [[biner]]. Teori ini memberikan kerangka formal untuk mengungkapkan pernyataan-pernyataan seperti "ini lebih kecil dari itu" atau "ini mendahului itu". Dalam artikel ini diperkenalkan bidang ini dan memberikan definisi dasar.
== Latar belakang dan motivasi ==
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=== Himpunan dengan tatanan parsial ===
Tatanan merupakan [[relasi biner]] khusus. Misalkan ''P'' adalah suatu himpunan dan ≤ adalah relasi terhadap ''P'', maka ≤ merupakan "tatanan parsial" (''partial order'') jika bersifat [[:en:reflexive relation|refleksif]], [[:en:antisymmetric relation|antisimetri]], dan [[:en:transitive relation|transitif]], yaitu untuk setiap ''a'', ''b'' dan ''c'' dalam ''P'', didapatkan:
:''a'' ≤ ''a'' (refleksivitas)
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:''a'' ≤ ''b'' atau ''b'' ≤ ''a'' (totalitas).
Tatanan-tatanan ini dapat juga disebut '''tatanan linear''' (''linear order'') atau '''rantai''' (''chain''). Banyak tatanan klasik bersifat [[Lincoln Near-Earth Asteroid Research|linear]], tetapi tatanan [[subset]] pada himpunan memberi contoh kapan hal ini tidak benar. Contoh lain dapat diberikan dari relasi divisibilitas "|". Untuk dua bilangan asli ''n'' dan ''m'', ditulis ''n''|''m'' jika ''n'' [[pembagian|dibagi oleh]] ''m'' tanpa sisa. Dapat dengan mudah dilihat bahwa ini menghasilkan tatanan parsial.<!--
The identity relation = on any set is also a partial order in which every two distinct elements are incomparable. It is also the only relation that is both a partial order and an [[equivalence relation]]. Many advanced properties of posets are interesting mainly for non-linear orders.
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=== Elemen khusus dalam suatu tatanan ===
Dalam suatu himpunan dengan tatanan parsial ada sejumlah [[elemen (matematika)|elemen]] yang berperan penting. Contoh paling dasar adalah "elemen terkecil" dalam suatu [[poset]]. Misalnya, 1 adalah elemen terkecil dari bilangan bulat positif dan [[himpunan kosong]] adalah himpunan terkecil di bawah tatanan subset. Secara formal, suatu elemen ''m''
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Notasi 0 sering dijumpai pada elemen terkecil, meskipun tidak melibatkan bilangan apapun. Namun, dalam tatanan suatu himpunan bilangan, notasi ini tidak tepat dan bahkan menimbulkan kerancuan, karena bilangan 0 tidak selalu yang terkecil. Contohnya adalah pada tatanan divisibilitas |, di mana 1 adalah elemen terkecil karena bilangan itu membangi semua bilangan yang lain. Sebaliknya, bilangan 0 merupakan bilangan yang dapat dibagi oleh semua bilangan lain. Jadi bilangan 0 merupakan '''elemen terbesar''' dari tatanan tersebut. Istilah lain untuk "terkecil" dan "terbesar" adalah "terendah" ("terbawah", "paling dasar"; ''bottom'') dan "tertinggi" ("teratas"; ''top'') dan juga "nol" (''zero'') dan "unit" ("satuan").
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<!--Least and [[greatest element]]s may fail to exist, as the example of the real numbers shows. But if they exist, they are always unique. In contrast, consider the divisibility relation | on the set {2,3,4,5,6}. Although this set has neither top nor bottom, the elements 2, 3, and 5 have no elements below them, while 4, 5, and 6 have none above. Such elements are called '''minimal''' and '''maximal''', respectively. Formally, an element ''m'' is [[minimal element|minimal]] if:
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: ''s'' ≤ ''b'', for all ''s'' in ''S''.
Lower bounds again are defined by inverting the order. For example, -5 is a lower bound of the natural numbers as a subset of the integers. Given a set of sets, an upper bound for these sets under the subset ordering is given by their [[union (set theory)|union]]. In fact, this upper bound is quite special: it is the smallest set that contains all of the sets. Hence, we have found the '''[[least upper bound]]''' of a set of sets. This concept is also called '''supremum''' or '''join''', and for a set ''S'' one writes sup(''S'') or v''S'' for its least upper bound. Conversely, the '''[[greatest lower bound]]''' is known as '''[[infimum]]''' or '''meet''' and denoted inf(''S'') or ^''S''. These concepts play an important role in many applications of order theory. For two elements ''x'' and ''y'', one also writes ''x'' v ''y'' and ''x'' ^ ''y'' for sup({''x'',''y''}) and inf({''x'',''y''}), respectively.
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For another example, consider again the relation | on natural numbers. The least upper bound of two numbers is the smallest number that is divided by both of them, i.e. the [[least common multiple]] of the numbers. Greatest lower bounds in turn are given by the [[greatest common divisor]].
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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
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.
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* [[Complete lattice]]s, where every set has a supremum and infimum, and
* [[Directed complete partial order]]s (dcpos), that guarantee the existence of suprema of all [[directed set|directed subsets]] and that are studied in [[domain theory]].
* [[Partial orders with complements]], or ''poc sets'',<ref>Martin A. Roller. Poc sets, median algebras and group actions. An extended study of Dunwoody's construction and Sageev's theorem. preprint, 1998.</ref> are posets ''S'' having a unique bottom element ''0∈S'', along with an order-reversing involution, such that <math>a\leq a^{*} \Rightarrow a=0</math>.
However, one can go even further: if all finite non-empty infima exist, then ∧ can be viewed as a total binary operation in the sense of [[universal algebra]]. Hence, in a lattice, two operations ∧ and ∨ are available, and one can define new properties by giving identities, such as
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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 ==
▲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]]
* [[Himpunan kausal]]
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* {{cite book|first=Garrett|last=Birkhoff|author-link=Garrett Birkhoff|title=Lattice Theory|volume=25|publisher=American Mathematical Society|edition=3rd Revised|year=1940|isbn=978-0-8218-1025-5}}
* {{cite book|first1=S. N.|last1=Burris|first2=H. P.|last2=Sankappanavar|year=1981|url=http://www.thoralf.uwaterloo.ca/htdocs/ualg.html|title=A Course in Universal Algebra|publisher= Springer|isbn=978-0-387-90578-5}}
* {{cite book|first1=B. A.|last1=Davey|first2=H. A.|last2=Priestley|year=2002|author2-link=Hilary Priestley|
* {{cite book|first1=G.|last1=Gierz|first2=K. H.|last2=Hofmann|first3=K.|last3=Keimel|first4=M.|last4=Mislove|first5=D. S.|last5=Scott|year = 2003|title=Continuous Lattices and Domains|publisher=Cambridge University Press|isbn=978-0-521-80338-0|series=Encyclopedia of Mathematics and its Applications|volume=93}}
== Pranala luar ==
{{Wiktionary|ordering}}
* [http://www.apronus.com/provenmath/orders.htm Orders at ProvenMath] partial order, linear order, well order, initial segment; formal definitions and proofs within the axioms of set theory.
* Nagel, Felix (2013). [http://www.felixnagel.org Set Theory and Topology. An Introduction to the Foundations of Analysis]
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