2.2: Orderings
A relation on a set may be thought of as part of the structure imposed on the set. Among the most important relations on a set are order relations.
Let
\(X\)
be a set and
\(R\)
a relation on
\(X\)
. We say that
\(R\)
is a partial ordering if:
(1)
\(R\)
is reflexive
(2)
\(R\)
is antisymmetric
(3)
\(R\)
is transitive.
Let \(X\) be a family of sets. The relation \(\subseteq\) is a partial ordering on \(X\) . Every set is a subset of itself, so the relation is reflexive. If \(Y \subseteq Z\) and \(Z \subseteq Y\) , then \(Y=Z\) , so the relation is antisymmetric. Finally, if \(Y \subseteq Z\) and \(Z \subseteq W\) then \(Y \subseteq W\) , so the relation is transitive.
Let \(R\) be the relation on \(\mathbb{N}^{+}\) defined by \(x R y\) if and only if there is \(z \in \mathbb{N}^{+}\) such that \[x z=y .\] Then \(R\) is a partial ordering of \(\mathbb{N}^{+}\) . (Prove this: Exercise 2.2).
Let \(X\) be a set and \(R\) be a partial ordering of \(X\) . We say that \(R\) is a linear ordering, also called a total ordering, provided that, for any \(x, y \in X\) , either \(x R y\) or \(y R x\) .
Note that since a linear ordering is antisymmetric, for any distinct \(x\) and \(y\) , exactly one of \(x R y\) and \(y R x\) holds.
The ordering \(\leq\) on \(\mathbb{N}\) (or \(\mathbb{R}\) ) is a linear ordering. So is the relation \(\geq\) . The relation \(<\) is not (why?).
Let \(X=\mathbb{R}^{n}\) . We can define a reflexive relation on \(X\) as follows. Let \(x=\left(a_{1}, \ldots, a_{n}\right)\) and \(y=\left(b_{1}, \ldots, b_{n}\right)\) be distinct members of \(X\) . Let \(k \in \mathbb{N}^{+}\) be the least number such that \(a_{k} \neq b_{k}\) . Then we define \[x R y \text { if and only if } a_{k}<b_{k} .\] Then \(R\) is a linear ordering of \(X\) . It is called the dictionary ordering.
The notion of a linear ordering is probably natural for you, and you have used it intuitively since you began studying arithmetic. The relation \(\leq\) helps you to visualize the set as a line in which the relative location of two elements of the set is determined by the linear ordering. If you are considering a set with operations, this in turn can help in visualizing how operations behave. For instance, think of using a number line to visualize addition, subtraction and multiplication of integers.
Partial orderings are generalizations of linear orderings, and \(\leq\) is the most obvious example of a linear ordering. Because of this, the normal symbol for a partial ordering is \(\preceq\) (it is also reminiscent of the symbol \(\subseteq\) , which is the example most mathematicians keep in mind when thinking about a partial ordering).
Let \(X\) be the set of all collections of apples and oranges. If \(x, y\) are in \(X\) , then say \(x \preceq y\) if the number of apples in \(x\) is less than or equal to the number of apples in \(y\) , and the number of oranges in \(x\) is less than or equal to the number of oranges in \(y\) . This is a partial ordering. You may not be able to compare apples to oranges, but you can say that 2 apples and 5 oranges is inferior to 4 apples and 6 oranges!
One way to visualize a partial order \(\preceq\) on a finite set \(X\) is to imagine arrows connecting distinct elements of \(X, x\) and \(y\) , if \(x \preceq y\) and there is no third distinct point \(z\) satisfying \(x \preceq z \preceq y\) . Then two elements \(a\) and \(b\) in \(X\) will satisfy \(a \preceq b\) if and only if you can get from \(a\) to \(b\) by following a path of arrows.
Consider the graph on the set \(X=\{a, b, c, d, e, f\}\) give in Figure 2.11.
It illustrates the partial order that could be described as the smallest reflexive, transitive relation \(\preceq\) on \(X\) that satisfies \(a \preceq b, a \preceq c, b \preceq\) \(d, b \preceq e, c \preceq e, e \preceq f .\)