Definition. A relation R on a set A is called a partial ordering or a - - PDF document

Definition. A relation R on a set A is called a partial ordering or a partial

• rder if it is reflexive, antisymmetric and transitive.

Then we denote this relation and say that the pair (A, ) is a partially ordered set, often poset for short.

Fact. Let (A, ) be a partially ordered set. Then the restriction to an arbitrary subset of A is also a partial ordering.

Fact. If (A, ) is a partially ordered set, then (A, −1) is also a poset.

Algorithm for deducing a Hasse diagram of a partially ordered set (A, ) for A finite.

• 1. Find elements a ∈ A that are never on the right in (sharp) relation

pairs, that is, in position x a (arrows do not end in them). Arrange them in the bottom row. Remove these elements from A and remove all relation pairs involving these elements.

• 2. In the remaining set find elements that are never on the right in

relation pairs. Arrange them into a second row counting from the

• bottom. Connect them with the points in the first row whenever

there is a relation there. Remove these elements from A, remove relation pairs involving these elements.

• 3. In the remaining set find elements that are never on the right

in relation pairs. Arrange them into a new row. Connect them with the points in the previous row whenever there is a relation

• there. Connect them with lower rows whenever there is a relation

there and there is no upwards path made of already existing edges. Remove these elements from A, remove relation pairs involving these elements. Repeat this step until no elements are left.

Definition. Let (A, ) be a partially ordered set and ≺ the corresponding derived

• relation. Let M be a non-empty subset of A.

We say that m ∈ A is the least element of M if m ∈ M and m x for all x ∈ M. We say that m ∈ A is the greatest element of M if m ∈ M and x m for all x ∈ M. We say that m ∈ A is minimal in M or a minimum of M, denoted m = min(M), if m ∈ M and there is no x ∈ M such that x ≺ m. We say that m ∈ A is maximal in M or a maximum of M, denoted m = max(M), if m ∈ M and there is no x ∈ M such that m ≺ x.

Theorem. Let (A, ) be a partially ordered set, consider a non-empty subset M ⊆ A. Then the following are true: (i) If there exists a least element of M, then it is unique. If there exists a greatest element of M, then it is unique. (ii) If m1 = min(M), m2 = min(M) and m1 m2, then m1 = m2. If m1 = max(M), m2 = max(M) and m1 m2, then m1 = m2. (iii) If m is the least element of M then m = min(M) and there is no other minimum of M. If m is the greatest element of M then m = max(M) and there is no other maximum of M.

Theorem. Let (A, ) be a partially ordered set. If M is a finite non-empty subset of A, then min(M) and max(M) exist.

Definition. Let (A, ) be a partially ordered set. We say that a, b ∈ A are comparable if a b or b a.

Definition. Let (A, ) be a partially ordered set. We say that is a total order or a linear order if every two elements of A are comparable.

Theorem. Let (A, ) be a linearly ordered set. If M is its non-empty finite subset, then it has the least and the greatest element.

Theorem. Let (A, ) be a finite partially ordered set. It is a linear order if and only if the elements of A can be written as A = {a1, . . . , an} in such a a way that a1 ≺ a2 ≺ · · · ≺ an.

Definition. Let (A, ) be a partially ordered set. A relation L on A is called a linear extension of if (A, L) is a linear order and ⊆L, that is, for all a, b ∈ A satisfying a b we also have a L b.

Theorem. For every finite linearly ordered set (A, ) there exists a linear ex- tension L on A.

procedure topological sort((A, )) k := 0; while A = ∅ do k := k + 1 ak := min(A) A := A − {ak};

• utput: (a1 ≺L a2 ≺L · · · ≺L ak);

Definition. Let (A, ) be a partially ordered set. We say that (A, ) is a well-ordered set if every non-empty subset

• f A has a least element.

Fact. Every well-ordering is also a linear ordering.

Axiom (well-ordering principle) (N, ≤) is a well-ordered set.

Definition. Consider partially ordered sets (A1, 1), . . . , (An, n). We define the lexicographic ordering on A = A1 × · · · × An as follows: For a = (a1, . . . , an), b = (b1, . . . , bn) ∈ A we set a L b exactly if ai = bi for all i = 1, . . . , n (that is, if a = b) or there exists and index k such that ai = bi for all i satisfying 1 ≤ i < k and ak ≺k bk.

Theorem. Consider well-ordered sets (A1, 1), . . . , (An, n). Then A = A1 × · · · × An equipped with the lexicographic ordering L is a well-

• rdered set.