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Partially Ordered Sets Dharmendra Mishra Partially Ordered Sets A relation R on a set A is called a partial order if R is reflexive, antisymmetric, and transitive. The set A together with the partial order R is called a partially ordered set , or simply a poset , and we will denote this poset by (A, R). If there is no possibility of confusion about the partial order, we may refer to the poset simply as A, rather than (A, R). EXAMPLE 1 Let A be a collection of subsets of a set S. The relation ⊆ of set inclusion is a partial order on A, so (A, ⊆) is a poset. EXAMPLE 2 Let Z + be the set of positive integers. The usual relation ≤ less than or equal to is a partial order on Z +, as is ≥ (grater than or equal to). EXAMPLE 3 The relation of divisibility (a R b is and only if a b )is a partial order on Z +. EXAMPLE 4 Let R be the set of all equivalence relations on a set A. Since R consists of subsets of A x A, R is a partially ordered set under the partial order of set containment. If R and S are equivalence relations on A, the same property may be expressed in relational notation as follows. R ⊆ S if and only if x R y implies x S y for all X, y in A. Than (R, ⊆) is poset. EXAMPLE 5 The relation < on Z + is not a partial order, since it is not reflexive. EXAMPLE 6 Let R be partial order on a set A, and let R -1 be the inverse relation of R. Then R -1 is also a partial order. To see this, we recall the characterization of reflexive, antisymmetric, and transitive given in section 4.4. If R has these three properties, then ∆ ⊆ R, R ∩ R -1 ⊆ ∆ , and R2 ⊆ R. By taking inverses. we have ∆ = ∆-1 ⊆ R -1, R -1 ∩ (R -1) = R -1 ∩ R ⊆ ∆, and (R -1)2 ⊆ R -1. so, by section 4.4, R -1 is relative, antisymmetric, and transitive. Thus R-1 is also a partial order. The poset (A, R -1) is called the dual of the poset (A, R) and the partial order R -1 is called the dual of the partial order R. The most familiar partial orders are the relations ≤ and ≥ on Z and R. For this reason, when speaking in general of a partial order R on a set A, we shall often use the symbols ≤ or ≥ for R. This makes the properties of R more familiar and easier to remember. Thus the reader may see the symbol ≤ used for many different partial orders on different sets. Do not mistake this to mean that these relations are all the same or that they are the familiar relation ≤ on Z or R. If it becomes absolutely necessary to distinguish partial orders from one another, we may also use symbols such as ≤1, ≤’, ≥1, ≥’, and so on to denote partial orders. We will observe the following convention. Whenever (A, ≤) is a poset, we will always use the symbol > for the partial order <-1, and thus (A, >) will be the duel poset. Similarly, the dual of poset (A, <1) Of the poset will be denoted by (A, >1) and the dual of the poset (B, <’)will be 1/90 Dharmendra Mishra denoted by (B,>’). Again, this convention is to remind us of the familiar duel poset. (Z, <) and (Z, >) as well as the posets (R, <) and (R, >). If (A, ≤) is a poset, the elements a and b of A are said to be comparable if a ≤ b or b ≤ a. Observe that in a partially ordered set every paid of elements need not be comparable. For example, consider the poset in Example 3. The elements 2 and 7 are not comparable, since 2 + 7 and 7 + 2. Thus the word “partial” in partially ordered set means that some elements may not be comparable. If every pair of elements in a poset A is comparable, we say that A is a linealy ordered set, and the partial order is called a linear order . We also say that A is a chain . EXAMPLE 7 The poset of Example 2 is linearly ordered. The following theorem is sometimes useful since it shows how to construct a new poset from given posets. Theorem 1 If (A, ≤) and (B, ≤) are posets, then (A X B, ≤ ) is a poset, with partial order ≤ defined by (a, b) ≤ (a’, b’) if a ≤ a’ in A and b ≤ b’ in B. Note that the symbol ≤ is being used to denote three distinct partial orders. The reader should find it easy to determine which of the three is meant at any time. Proof If (a, b) ∈ A X B, then (a, b) ≤ (a, b) since a ≤ a in A and b ≤ b in B, so ≤ satisfies the reflexive property in A X B. Now suppose that (a, b) ≤ (a’, b’) and (a’, b’) ≤ (a, b) where a and a’ A ∈ and b and b’ ∈ a B. Then a ≤ a’ and a’ ≤ a in A and b ≤ b’ and b’ ≤ b in B. Since A and B are posets, the antisymmetry of the partial orders on A and B implies that a = a’ and b = b’. Hence ≤ satisfies the antisymmetry property in A X B. Finally, suppose that (a, b) ≤ (a’, b’) and (a’, b’) ≤ (a’’, b’’). where a, a’, a’’ ∈ A, and b, b’, b’’ ∈ B. Then a ≤ a’ and a’ ≤ a’’, so a ≤ a’’, by the transitive property of the partial order on A. Similarly, b ≤ b’ and b’’ ≤ b’’. 2/90 Dharmendra Mishra so b ≤ b’’, by the transitive property of the partial order on B. Hence (a, b) ≤ (a’, b’’). Consequently, the transitive property holds for the partial order on A X B, and wee conclude that A X B is a poset. The partial order < defined on the Cartesian product A X B is called the product partial order. If (A, < ) is a poset, we say that a < b if a < b but a ≠ b. Suppose now that (A, <) and (B, < ) are posets. In Theorem 1 we have defined the product partial order on A X B. Another useful partial order on A*B, denoted by <, is defined as follows. (a, b) < (a’, b’) if a < a’ or if a = a’ and b < b’. This ordering is called lexicographic, or “dictionary” order. The ordering of the elements in the first coordinate, expect in case of “ties,” when attention passes to the second coordinate. If (A, <) and (B, < ) are linearly ordered sets, then the lexicographic order < on A X B is also a linear order. EXAMPLE 8 Let A = R, with usual ordering <. Then the plane R 2 = R X R may be fiven lexicographic order. This is illustrated in Figure 1. We see that the plane is linearly ordered by lexicographic order. Each vertical line has the usual order, and points on one line are less than points on a line farther to the right. Thus, in Figure 6.1, P 1 < P 2, P 1 < P 3 and P 2 < P 3. Lexicographic ordering is easily extended to Cartesian products A1 X A 2 X…X A n as follows: (a 1, a 2,….a n) < (a’ 1, a’ 2….a’ n) if and only if a1 < a’ 1 or a1 = a’ 1 and a 2 < a’ 2 or a1 = a’ 1, a 2 = a’ 2 and a 3 < a’ 3 or … a1 = a’ 1 a 2 = a’ 2 …. A n-1 = a’ n-1 and a n < a’ n. Thus the first coordinate dominates expect for equality, in which case we consider the second coordinate. If equality holds again, we pass to the next coordinate, and so on. EXAMPLE 9 Let S = (a, b,…..z) be the ordinary alphabet, linearly ordered in the usual way (a < b, b < c…..y < z). Then S n = S X S X…X S (n factors) can be identified with the set of all words having length n. Lexicographic order on S n dictionary listing. This fact accounts for the name of the ordering. Thus park < part, help < hind, jump < mump. The third is true since j < m; the second since h =, e < i; and the first is true since p = p, a= a, r = r, k < t. If S is a poset, we can extend lexicographic order to S* (see Section 1.3) in the following way. If x = a 1a2…a n and y = b 1b2…b k are in S* with n < k, we say that x < y if (a 1….a n) < (b 1…..b n) in S n under lexicographic ordering of S n. 3/90 Dharmendra Mishra In the previous paragraph, we use the fact that the n-tuple (a 1, a 2….a n) ∈ S n, and the string a 1a2…a n ∈ S* are really the same sequence of length n, written in two different notations. The notations differ for historical reasons, and we will use them interchangeably depending on context. EXAMPLE 10 Let S be (a…..z), ordered as usual. Then S* is the set of all possible “words” of any length, whether such words are meaningful or not. Thus we have Help < helping In S* SINCE Help < help In S 4. Similarly, we have Help < helping Since Help < helping In S 6. As the example Help < helping Shows, this order includes prefix order that is, any word is greater than all of its prefixes (beginning parts). This is also the way that words occur in the dictionary.
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