Teori order: Perbedaan antara revisi

'''Teori order''' ({{lang-en|order theory}}) atau '''teori tatanan''' (= 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. <!--A list of order-theoretic terms can be found in the [[order theory glossary]].-->

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''&nbsp;v&nbsp;''y'' and ''x''&nbsp;^&nbsp;''y'' for sup({''x'',''y''}) and inf({''x'',''y''}), respectively. <!-- Using Wikipedia's [[meta:MediaWiki User's Guide: Editing mathematical formulae|TeX markup]], one can also write <math>\vee</math> and <math>\wedge</math>, as well as the larger symbols <math>\bigvee</math> and <math>\bigwedge</math>. Note however, that all of these symbols may fail to match the font size of the standard text and should therefore preferably be used in extra lines. The rendering of ∨ for v and ∧ for ^ is not supported by many of today's [[web browser]]s across all platforms and therefore avoided here.-->