Partially ordered set
In
mathematics, especially order theory, a partially ordered set is a
set equipped with a partial order relation. This relation formalizes the intuitive concept of an ordering, sequencing, or arrangement of that set's elements. Such an ordering does not necessarily need to be total, that is, it need not guarantee the mutual comparability of all objects in the set, but it can be. A poset defines a poset topology.
Encyclopedia
In
mathematics, especially order theory, a
partially ordered set is a
set equipped with a partial order relation. This relation formalizes the intuitive concept of an ordering, sequencing, or arrangement of that set's elements. Such an ordering does not necessarily need to be total, that is, it need not guarantee the mutual comparability of all objects in the set, but it can be. A poset defines a poset topology.
Formal definition
A
partial order is a binary relation
R over a
set P which is
reflexive,
antisymmetric, and transitive, i.e., for all
a,
b, and
c in
P, we have that:
- aRa ;
- if aRb and bRa then a = b ; and
- if aRb and bRc then aRc .
A set with a partial order is called a
partially ordered set. 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.
Examples
...
relation .
- The set of natural numbers equipped with the divides relation.
Strict and weak partial orders
In some contexts, the partial order defined above is called a
weak partial order. In these contexts a
strict partial order is a binary relation that is
irreflexive and transitive, and therefore
antisymmetric. In other words, for all
a,
b, and
c in
P, we have that:
- ¬ ;
- if a ≠ b and aRb then ¬ ; and
- if aRb and bRc then aRc .
If
R is a weak partial order, then
R − is the corresponding strict partial order. Similarly, every strict partial order has a corresponding weak partial order, and so the definition of each is readily expressed in terms of the other.
Strict partial orders are also useful because they correspond more directly to
directed acyclic graphs : 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.
See also:
strict weak orderingWhen considered as a category where hom = and
o = , posets are equivalent to one another if and only if they are isomorphic. In a poset, the smallest element, if any, is an initial object, and the largest element, if any, a terminal object. Also, every pre-ordered set is equivalent to a poset. Finally, every subcategory of a poset is isomorphism-closed.
See also
- order theory
- preorder
- total order
- directed set
- equivalence relation
- Hasse diagram
- graded poset
