Editing Glossary of order theory (section)

== D ==

* '''dcpo'''. See ''[[directed complete partial order]]''.
* '''[[Dedekind–MacNeille completion]]'''. The Dedekind–MacNeille completion of a [[partially ordered set]] is the smallest [[complete lattice]] that contains it.
* '''[[Dense order]]'''. A '''[[Dense order|dense]]''' poset ''P'' is one in which, for all elements ''x'' and ''y'' in ''P'' with ''x'' < ''y'', there is an element ''z'' in ''P'', such that ''x'' < ''z'' < ''y''.  A subset ''Q'' of ''P'' is '''dense in''' ''P'' if for any elements ''x'' < ''y'' in ''P'', there is an element ''z'' in ''Q'' such that ''x'' < ''z'' < ''y''.
* '''[[Derangement]]'''. A permutation of the elements of a set, such that no element appears in its original position. 
* '''[[Directed set]]'''. A [[non-empty]] subset ''X'' of a poset ''P'' is called directed, if, for all elements ''x'' and ''y'' of ''X'', there is an element ''z'' of ''X'' such that ''x'' ≤ ''z'' and ''y'' ≤ ''z''. The dual notion is called ''filtered''.
* '''[[Directed complete partial order]]'''. A poset ''D'' is said to be a directed complete poset, or '''dcpo''', if every directed subset of ''D'' has a supremum.
* '''[[Distributivity (order theory)|Distributive]]'''. A lattice ''L'' is called distributive if, for all ''x'', ''y'', and ''z'' in ''L'', we find that ''x'' &and; (''y'' &or; ''z'') = (''x'' &and; ''y'') &or; (''x'' &and; ''z''). This condition is known to be equivalent to its order dual. A meet-[[semilattice]] is distributive if for all elements ''a'', ''b'' and ''x'', ''a'' &and; ''b'' ≤ ''x'' implies the existence of elements ''a' '' ≥ ''a'' and ''b' '' ≥ ''b'' such that ''a' '' &and; ''b' '' = ''x''. See also ''completely distributive''.
* '''[[Domain theory|Domain]]'''. Domain is a general term for objects like those that are studied in [[domain theory]]. If used, it requires further definition.
* '''Down-set'''. See ''lower set''.
* '''[[Duality (order theory)|Dual]]'''. For a poset (''P'', ≤), the dual order ''P''<sup>''d''</sup> = (''P'', ≥) is defined by setting ''x ≥ y'' [[if and only if]] ''y ≤ x''. The dual order of ''P'' is sometimes denoted by ''P''<sup>op</sup>, and is also called ''opposite'' or ''converse'' order. Any order theoretic notion induces a dual notion, defined by applying the original statement to the order dual of a given set. This exchanges ≤ and ≥, meets and joins, zero and unit.