Editing Glossary of order theory (section)

== C ==

* '''[[Total order#Chains|Chain]]'''. A chain is a totally ordered set or a totally ordered subset of a poset. See also {{em|total order}}.
* '''[[Chain complete]]'''.  A [[partially ordered set]] in which every [[Total order#Chains|chain]] has a [[least upper bound]].
* '''[[Closure operator]]'''. A closure operator on the poset ''P'' is a function ''C'' : ''P'' → ''P'' that is monotone, [[idempotent]], and satisfies ''C''(''x'') ≥ ''x'' for all ''x'' in ''P''.
* '''[[Compact element|Compact]]'''. An element ''x'' of a poset is compact if it is ''[[Way-below relation|way below]]'' itself, i.e. ''x''<<''x''. One also says that such an ''x'' is {{em|finite}}.
* '''[[Comparability|Comparable]]'''. Two elements ''x'' and ''y'' of a poset ''P'' are comparable if either ''x'' ≤ ''y'' or ''y'' ≤ ''x''.
* '''[[Comparability graph]]'''.  The comparability graph of a poset (''P'', ≤) is the [[Graph (discrete mathematics)|graph]] with vertex set ''P'' in which the edges are those pairs of distinct elements of ''P'' that are comparable under ≤ (and, in particular, under its reflexive reduction <).
* '''[[Complete Boolean algebra]]'''.  A [[Boolean algebra (structure)|Boolean algebra]] that is a complete lattice.
* '''[[Complete Heyting algebra]]'''. A [[Heyting algebra]] that is a complete lattice is called a complete Heyting algebra. This notion coincides with the concepts ''frame'' and ''locale''.
* '''[[Complete lattice]]'''. A complete [[Lattice (order)|lattice]] is a poset in which arbitrary (possibly infinite) joins (suprema) and meets (infima) exist.
* '''[[Complete partial order]]'''. A complete partial order, or '''cpo''', is a [[directed complete partial order]] (q.v.) with least element.
* '''Complete relation'''.  Synonym for ''[[Connected relation]]''.
* '''Complete semilattice'''. The notion of a ''complete semilattice'' is defined in different ways. As explained in the article on [[completeness (order theory)]], any poset for which either all suprema or all infima exist is already a complete lattice. Hence the notion of a complete semilattice is sometimes used to coincide with the one of a complete lattice. In other cases, complete (meet-) semilattices are defined to be [[bounded complete]] [[complete partial order|cpos]], which is arguably the most complete class of posets that are not already complete lattices.
* '''[[Completely distributive lattice]]'''. A complete lattice is completely distributive if arbitrary joins distribute over arbitrary meets.
* '''Completion'''. A completion of a poset is an [[order-embedding]] of the poset in a complete lattice.
* '''[[Dedekind–MacNeille completion|Completion by cuts]]'''. Synonym of [[Dedekind–MacNeille completion]].
* '''[[Connected relation]]'''.  A total or complete relation ''R'' on a set ''X'' has the property that for all elements ''x'', ''y'' of ''X'', at least one of ''x R y'' or ''y R x'' holds.
* '''[[Continuous poset]]'''. A poset is continuous if it has a '''base''', i.e. a subset ''B'' of ''P'' such that every element ''x'' of ''P'' is the supremum of a directed set contained in {''y'' in ''B'' | ''y''<<''x''}.
* '''Continuous function'''. See ''Scott-continuous''.
* '''Converse'''.  The converse <° of an order < is that in which x <° y whenever y < x.
* '''Cover'''. An element ''y'' of a poset ''P'' is said to cover an element ''x'' of ''P'' (and is called a cover of ''x'') if ''x'' < ''y'' and there is no element ''z'' of ''P'' such that ''x'' < ''z'' < ''y''.
* '''[[Complete partial order|cpo]]'''. See ''complete partial order''.