http://debianws.lexgopc.com/wiki143/index.php?action=history&feed=atom&title=Preordered_class 2026-10-26T03:16:33Z Revision history for this page on the wiki MediaWiki 1.43.1 http://debianws.lexgopc.com/wiki143/index.php?title=Preordered_class&diff=6124708&oldid=prev 2022-03-19T11:45:08Z

<p></p> <p><b>New page</b></p><div>{{Short description|Class equipped with a preorder}}<br /> In [[mathematics]], a &#039;&#039;&#039;preordered class&#039;&#039;&#039; is a [[class (mathematics)|class]] equipped with a [[preorder]].<br /> <br /> ==Definition==<br /> When dealing with a class &#039;&#039;C&#039;&#039;, it is possible to define a class relation on &#039;&#039;C&#039;&#039; as a [[Subclass (set theory)|subclass]] of the power class &#039;&#039;C &lt;math&gt; \times &lt;/math&gt; C&#039;&#039; . Then, it is convenient to use the language of [[relation (mathematics)|relations]] on a set.<br /> <br /> A &#039;&#039;&#039;preordered class&#039;&#039;&#039; is a class with a [[preorder]] on it. &#039;&#039;Partially ordered class&#039;&#039; and &#039;&#039;totally ordered class&#039;&#039; are defined in a similar way. These concepts generalize respectively those of [[preordered set]], [[partially ordered set]] and [[totally ordered set]]. However, it is difficult to work with them as in the &#039;&#039;small&#039;&#039; case because many constructions common in a [[set theory]] are no longer possible in this framework.<br /> <br /> Equivalently, a preordered class is a &#039;&#039;&#039;[[thin category]]&#039;&#039;&#039;, that is, a [[category (mathematics)|category]] with at most one morphism from an object to another.<br /> <br /> ==Examples==<br /> *In any [[category (mathematics)|category]] &#039;&#039;C&#039;&#039;, when &#039;&#039;D&#039;&#039; is a class of morphisms of &#039;&#039;C&#039;&#039; containing identities and closed under composition, the relation &#039;there exists a &#039;&#039;D&#039;&#039;-morphism from &#039;&#039;X&#039;&#039; to &#039;&#039;Y&#039;&#039;&#039; is a preorder on the class of objects of &#039;&#039;C&#039;&#039;.<br /> *The class &#039;&#039;&#039;Ord&#039;&#039;&#039; of all [[ordinal number|ordinal]]s is a totally ordered class with the classical ordering of ordinals.<br /> <br /> ==References==<br /> *Nicola Gambino and Peter Schuster, Spatiality for formal topologies<br /> *{{cite book | last = Adámek | first = Jiří |author2=Horst Herrlich |author3=George E. Strecker | year = 1990 | url = http://katmat.math.uni-bremen.de/acc/acc.pdf | title = Abstract and Concrete Categories | publisher = John Wiley &amp; Sons | isbn = 0-471-60922-6}}<br /> <br /> [[Category:Order theory]]<br /> [[Category:Set theory]]</div>

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