Glossary of Order Theory - P

P

• Partial order. A partial order is a binary relation that is reflexive, antisymmetric, and 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 (P, ≤), or poset for short, is a set P together with a partial order ≤ on P.

• Poset. A partially ordered set.

• Preorder. A preorder is a binary relation that is reflexive and transitive. Such orders may also be called quasiorders. The term preorder is also used to denote an acyclic binary relation (also called an acyclic digraph).

• 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.

• Prime. An ideal I in a lattice L is said to be prime, if, for all elements x and y in L, x ∧ 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.

• 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 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 ∧ x = 0}, i.e. as the least upper bound of all elements y with y ∧ x = 0.