Partial Orders vs Total Orders: An Example of a Partial Order That Is Not a Total Order

Understanding the concepts of partial orders and total orders is fundamental in the study of discrete mathematics and order theory. In this article, we will explore an important example that illustrates a partial order that is not a total order.

Definition of Partial Order and Total Order

A partial order on a set is a binary relation over the set that is reflexive, antisymmetric, and transitive. In contrast, a total order (also known as a linear order) is a special case of a partial order where every pair of elements is comparable.

Example: A Partial Order That Is Not a Total Order

Consider the set S {a, b, c} with the following relation R:

a is related to a (reflexivity) b is related to b (reflexivity) c is related to c (reflexivity) a is related to b (denoted a leq b) a is related to c (denoted a leq c) b and c are not related (denoted b leq c and c leq b)

The relation can be represented as follows:

a leq b a leq c b leq c and c leq b

Properties:

Reflexivity

Each element is related to itself. In our set, a, b, and c are all related to themselves.

Antisymmetry

If x leq y and y leq x then x y holds trivially since b and c are not comparable. Specifically, if b were related to c and vice versa, they would be equal.

Transitivity

If x leq y and y leq z, then x leq z for the pairs involving a. For example, since a leq b and b leq c, it follows that a leq c.

Conclusion

This relation is a partial order because it satisfies all three properties: reflexivity, antisymmetry, and transitivity. However, it is not a total order because b and c are not comparable. This means that there are elements in the set that do not have a direct comparison or order.

Additional Examples

The partial order of subsets of a set under the set inclusion order is generally not a total order. For instance, consider the set A consisting of all subsets of {a, b, c, d, e}. A can be ordered by the relation R defined by xRy if and only if x is a subset of y. This relation is transitive, reflexive, and antisymmetric. However, it is not a total order because neither {a, b, c} nor {c, d, e} is a subset of the other.

Similarly, if A is the set of natural numbers and R is the relation “is a factor of,” this defines a partial order. However, neither 6 nor 9 is a factor of the other, so this partial order is also not a total order.

By exploring these examples, we gain a deeper understanding of the differences between partial orders and total orders. Understanding these concepts is crucial for various applications in computer science, mathematics, and other fields where order theory plays a significant role.