The relation is called a partial order on . In practice, is usually conflated with ; if a distinction is needed, is called the ground set or underlying set of . The binary relation defined by removing the diagonal from (i.e. iff and ) satisfies the following properties:
is irreflexive, so if holds, then does not hold; and
Since is reflexive, it can be uniquely recovered from by adding the diagonal. For this reason, an irreflexive and transitive binary relation (called a strict partial order) also defines a poset, by means of the associated relation described above (which is called weak partial order).
Since every partial order is reflexive and transitive, every poset is a preorder. The notion of partial order is stricter than that of preorder, Let be the structure with ground set and binary relation . A diagram of this structure, omitting loops, is displayed below.
Observe that the binary relation on is reflexive and transitive, so is a preorder. On the other hand, and , while . So the binary relation on is not antisymmetric, implying that is not a poset.
Since every total order is reflexive, antisymmetric, and transitive, every total order is a poset. The notion of partial order is weaker than that of total order. A total order must obey the trichotomy law, which states that for any and in the order, either or . Let be the structure with ground set and binary relation . A diagram of this structure, omitting loops, is displayed below.
Observe that the binary relation on is reflexive, antisymmetric, and transitive, so is a poset. On the other hand, neither nor holds in . Thus fails to satisfy the trichotomy law and is not a total order.
The failure of the trichotomy law for posets motivates the following terminology. Let be a poset. If or holds in , we say that and are comparable; otherwise, we say they are incomparable. We use the notation to indicate that and are incomparable.
If and are posets, then a function is said to be order-preserving, or monotone, provided that it preserves inequalities. That is, is order-preserving if whenever holds, it follows that also holds. The identity function on the ground set of a poset is order-preserving. If , , and are posets and and are order-preserving functions, then the composition is also order-preserving.
Posets together with order-preserving functions form a category, which we denoted by . Thus an order-preserving function between the ground sets of two posets is sometimes also called a morphism of posets. The category of posets has arbitrary products (http://planetmath.org/ProductofPosets). Moreover, every poset can itself be viewed as a category, and it turns out that a morphism of posets is the same as a functor between the two posets.
Examples of posets
The two extreme posets are the chain, in which any two elements are comparable, and the antichain, in which no two elements are comparable. A poset with a singleton underlying set is necessarily both a chain and an antichain, but a poset with a larger underlying set cannot be both.
Let be the set of natural numbers. Inductively define a binary relation on by the following rules:
for any , the relation holds; and
whenever , the relation also holds.
Then is a chain, hence a poset. This structure can be naturally embedded in the larger chains of the integers, the rational numbers, and the real numbers.
The next example shows that nontrivial antichains exist.
So far the only posets we have seen are chains and antichains. Most posets are neither. The following construction gives many such examples.
There are important structure theorems for posets concerning chains and antichains. One of the foundational results is Dilworth’s theorem. This theorem was massively generalized by Greene and Kleitman.
A final example shows that one can manufacture a poset from an existing one.
Let be a poset ordered by . The dual poset of is defined as follows: it has the same underlying set as , whose order is defined by iff . It is easy to see that is a partial order. The dual of is usually denoted by .
Graph-theoretical view of posets
Let be a poset with strict partial order . Then can be viewed as a directed graph with vertex set the ground set of and edge set . For example, the following diagram displays the Boolean algebra as a directed graph.
If is a sufficiently complicated poset, then drawing all of the edges of can obscure rather than reveal the structure of . For this reason it is convenient to restrict attention to a subrelation of from which can be uniquely recovered.
We describe a method of constructing a canonical subgraph of from which the partial order can be recovered as long as every interval of has finite height. If and are elements of , then we say that covers if and there are no elements of strictly larger than but strictly smaller than , that is, if . Two elements are said to be consecutive if one covers another. Define a binary relation on by
Suppose every interval of has finite height. Then is the transitive closure of .
We prove this by induction on height. By definition of , if and the height of is 1, then .
Assume for induction that whenever and the height of is at most , then is in the transitive closure of . Suppose that and that the height of is . Since every chain in is finite, it contains an element which is strictly larger than and minimal (http://planetmath.org/MaximalElement) with respect to this property. Therefore , from which we conclude that . Since the interval is a proper subinterval of , it has height at most , so by the induction assumption we conclude that is in the transitive closure of . Since and are in the transitive closure of , so is . Hence whenever and the height of is at most , then is in the transitive closure of .
This completes the proof. ∎
For simplicity, the Hasse diagram of a poset is usually drawn as an undirected graph. Elements which are higher in the partial order relation are drawn physically higher. Since a strict partial order is acyclic, this can be done uniquely and the partial order can be recovered from the drawing.
|Date of creation||2013-03-22 11:43:41|
|Last modified on||2013-03-22 11:43:41|
|Last modified by||mps (409)|
|Synonym||partially ordered set|
|Defines||morphism of posets|