Editing Glossary of order theory (section)

== P ==

* '''[[Partial order]]'''. A partial order is a [[binary relation]] that is [[Reflexive relation|reflexive]], [[Antisymmetric relation|antisymmetric]], and [[Transitive relation|transitive]]. In a slight abuse of terminology, the term is sometimes also used to refer not to such a relation, but to its corresponding partially ordered set.
* '''[[Partially ordered set]]'''. A partially ordered set <math>(P, \leq),</math> or {{em|poset}} for short, is a set <math>P</math> together with a partial order <math>\,\leq\,</math> on <math>P.</math>
* '''Poset'''. A partially ordered set.
* '''[[Preorder]]'''. A preorder is a [[binary relation]] that is [[Reflexive relation|reflexive]] and [[Transitive relation|transitive]]. Such orders may also be called {{em|quasiorders}} or {{em|non-strict preorder}}. The term {{em|preorder}} is also used to denote an [[#A|acyclic]] [[binary relation]] (also called an {{em|acyclic digraph}}).
* '''[[Preordered set]]'''. A preordered set <math>(P, \leq)</math> is a set <math>P</math> together with a preorder <math>\,\leq\,</math> on <math>P.</math>
* '''Preserving'''. A function ''f'' between posets ''P'' and ''Q'' is said to preserve suprema (joins), if, for all subsets ''X'' of ''P'' that have a supremum sup ''X'' in ''P'', we find that sup{''f''(''x''): ''x'' in ''X''} exists and is equal to ''f''(sup ''X''). Such a function is also called '''join-preserving'''. Analogously, one says that ''f'' preserves finite, non-empty, directed, or arbitrary joins (or meets). The converse property is called ''join-reflecting''.
* '''[[Order ideal|Prime]]'''. An {{em|ideal}} ''I'' in a lattice ''L'' is said to be prime, if, for all elements ''x'' and ''y'' in ''L'', ''x'' &and; ''y'' in ''I'' implies ''x'' in ''I'' or ''y'' in ''I''. The dual notion is called a ''prime filter''. Equivalently, a set is a prime filter [[if and only if]] its complement is a prime ideal.
* '''[[Order ideal|Principal]]'''. A filter is called ''principal filter'' if it has a least element. Dually, a ''principal ideal'' is an ideal with a greatest element. The least or greatest elements may also be called ''principal elements'' in these situations.
* '''Projection (operator)'''. A self-map on a [[partially ordered set]] that is [[Monotonic function|monotone]] and [[idempotent]] under [[function composition]]. Projections play an important role in [[domain theory]].
* '''Pseudo-complement'''. In a [[Heyting algebra]], the element ''x'' ⇒; ''0'' is called the pseudo-complement of ''x''. It is also given by sup{''y'' :  ''y'' &and; ''x'' = 0}, i.e. as the least upper bound of all elements ''y'' with ''y'' &and; ''x'' = 0.