Partially ordered set

In mathematics, especially order theory, a partially ordered set (or poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. Such a relation is called a partial order to reflect the fact that not every pair of elements need be related: for some pairs, it may be that neither element precedes the other in the poset. Thus, partial orders generalize the more familiar total orders, in which every pair is related. A finite poset can be visualized through its Hasse diagram, which depicts the ordering relation.

A familiar real-life example of a partially ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs bear no such relationship.

A (non-strict) partial order is a binary relation ≤ over a set P which is reflexive, antisymmetric, and transitive, i.e., which satisfies for all a, b, and c in P:

In other words, a partial order is an antisymmetric preorder.

A set with a partial order is called a partially ordered set (also called a poset). The term ordered set is sometimes also used, as long as it is clear from the context that no other kind of order is 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.

For a, b, elements of a partially ordered set P, if a ≤ b or b ≤ a, then a and b are comparable. Otherwise they are incomparable. In the figure on top-right, e.g. {x} and {x,y,z} are comparable, while {x} and {y} are not. A partial order under which every pair of elements is comparable is called a total order or linear order; a totally ordered set is also called a chain (e.g., the natural numbers with their standard order). A subset of a poset in which no two distinct elements are comparable is called an antichain (e.g. the set of singletons {{x}, {y}, {z}} in the top-right figure). An element a is said to be covered by another element b, written a<:b, if a is strictly less than b and no third element c fits between them; formally: if both a≤b and a≠b are true, and a≤c≤b is false for each c with a≠c≠b. A more concise definition will be given below using the strict order corresponding to "≤". For example, {x} is covered by {x,z} in the top-right figure, but not by {x,y,z}.

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