Partial orderings

A relation R on a set S is called a partial ordering or partial order if it is reflexive, antisymmetric and transitive. A set S together with a partial ordering R is called a partially ordered set, or poset, and is denoted by (S, R).

Example. Show that the “greater than or equal” relation ≥ is a partial ordering on the set of integers.

Solution: Since for every integer a, ≥ is reflexive. If and , then a = b. Hence, ≥ is antisymmetric. Finally, ≥ is transitive since and imply that . It follows that ≥ is a partial ordering on the set of integers and is a poset.

Example. The divisibility relation | is a partial ordering on the set of positive integers, since it is reflexive, antisymmetric and transitive. We see that is a poset ( denotes the set of positive integers).

Example. Show that the inclusion relation  is a partial ordering on the power set of a set S.

Solution: Since whenever A is a subset of S,  is reflexive. It is antisymmetric since and imply that A = B. Finally,  is transitive, since and imply that . Hence,  is a partial ordering on P(S), and is a poset.

In a poset the notation denotes that . The notation denotes that , but . Also, we say “a is less than b” or “b is greater than a” if .

The elements a and b of a poset are called comparable if either or . When a and b are elements of S such that neither nor , a and b are called incomparable.

Example. In the poset , are the integers 3 and 9 comparable? Are 5 and 7 comparable?

Solution: The integers 3 and 9 are comparable, since 3 | 9. The integers 5 and 7 are incomparable, because and

The adjective “partial” is used to describe partial orderings since pairs of elements may be incomparable. When every two elements in the set are comparable, the relation is called a total ordering.

If is a poset and every two elements of S are comparable, S is called a totally ordered or linearly ordered set, and ≤ is called a total order or a linear order. A totally ordered set is also called a chain.

Example. The poset is totally ordered, since or whenever a and b are integers.

Example. The poset is not totally ordered since it contains elements that are incomparable, such as 5 and 7.

is a well-ordered set if it is a poset such that ≤ is a total ordering and such that every nonempty subset of S has a least element.

Example. The set of ordered pairs of positive integers, , with if , or if and (the lexicographic ordering), is a well-ordered set. The set Z, with the usual ≤ ordering, is not well-ordered since the set of negative integers, which is a subset of Z, has no least element.

Lexicographic order

The words in a dictionary are listed in alphabetic, or lexicographic, order, which is based on the ordering of the letters in the alphabet. This is a special case of an ordering of strings on a set constructed from a partial ordering on the set.

The lexicographic ordering on is defined by specifying that one pair is less than a second pair if the first entry of the first pair isles than (in A₁) the first entry of the second pair, or if the first entries are equal, but the second entry of this pair is less than (in A₂) the second entry of the second pair. In other words, , either if or if both and . We obtain a partial ordering by adding equality to the ordering on .

Example. Determine whether , whether , whether in the poset , where is the lexicographic ordering constructed from the usual ≤ relation on Z.

A lexicographic ordering can be defined on the Cartesian product of n posets , , …, . Define the partial ordering on by if , or if there is an integer such that and .

We can now define lexicographic ordering of strings. Consider the strings and on a partially ordered set S. Suppose these strings are not equal. Let t be the minimum of m and n. The definition of lexicographic ordering is that the string is less than iff or and m < n where in this inequality represents the lexicographic ordering of S ^t.

Example. Consider the set of strings of lowercase English letters. Using the ordering of letters in the alphabet, a lexicographic ordering on the set of strings can be constructed. A string is less than a second string if the letter in the first string in the first position where the strings differ comes before the letter in the second string in this position, or if the first string and the second string agree in all positions, but the second string has more letters. This ordering is the same as that used in dictionaries. For example, discreet discrete, since these strings differ first in the seventh position, and e t. Also, discreet discreetness, since the first eight letters agree, but the second string is longer. Furthermore, discrete discretion, since discrete discreti.