Sajun.org
In
mathematics, '''partially ordered sets''', or '''posets''' for short, are special
binary relations which formalize the intuitive concept of an ordering. Partially ordered sets are studied in
order theory and a much more detailed introduction to the field can be found within the corresponding article. In contrast, this article serves as a quick lookup for the formal definition.
== Formal definition ==
A
binary relation ''R'' over a
set ''P'' is a '''weak partial order''' if it is
reflexive,
antisymmetric, and
transitive, i.e., for all ''a'', ''b'' and ''c'' in ''P'', we have that:
:''aRa'' (reflexivity);
: if ''aRb'' and ''bRa'' then ''a'' = ''b'' (antisymmetry); and
: if ''aRb'' and ''bRc'' then ''aRc'' (transitivity).
Alternatively, a '''strict partial order''' is a binary relation which is
irreflexive,
asymmetric, and
transitive. In other words, for all ''a'', ''b'', and ''c'' in ''P'', we have that:
: ¬(''aRa'') (irreflexivity);
: if ''aRb'' then ¬(''bRa'') (asymmetry); and
: if ''aRb'' and ''bRc'' then ''aRc'' (transitivity).
If ''R'' is a weak partial order, then ''R'' − {(''a'', ''a'') | ''a'' in ''P''} is the corresponding strict partial order. Similarly, every strict partial order has a corresponding weak partial order, and so the two definitions are essentially equivalent.
In mathematics, '''''partial order''''' usually means ''weak partial order''. However, strict partial orders are also useful because they correspond more directly to
directed acyclic graphs (dags): every strict partial order is a dag, and the
transitive closure of a dag is both a strict partial order and also a dag itself.
A set with a partial order is called a '''partially ordered set''', or '''poset''' for short. The term '''ordered set''' is sometimes also used for posets, as long as it is clear from the context that no other kinds of orders are meant. In particular,
totally ordered sets can also be referred to as "ordered sets", especially in areas where these structures are more common than posets. However, most articles should not cause confusion as long as all formal definitions employ exact terminology.
== See also ==
*
Order theory
*
Preorder
*
Total order
'''Compare:'''
Equivalence class,
Directed set.
es:Conjunto parcialmente ordenado
