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 order 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 m 1 = min( M ), m 2 = min( M ) and m 1 � m 2 , then m 1 = m 2 . If m 1 = max( M ), m 2 = max( M ) and m 1 � m 2 , then m 1 = m 2 . (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 = { a 1 , . . . , a n } in such a a way that a 1 ≺ a 2 ≺ · · · ≺ a n .

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 a k := min( A ) A := A − { a k } ; output: ( a 1 ≺ L a 2 ≺ L · · · ≺ L a k ) ;

Definition. Let ( A, � ) be a partially ordered set. We say that ( A, � ) is a well-ordered set if every non-empty subset of 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 ( A 1 , � 1 ) , . . . , ( A n , � n ). We define the lexicographic ordering on A = A 1 × · · · × A n as follows: For a = ( a 1 , . . . , a n ) , b = ( b 1 , . . . , b n ) ∈ A we set a � L b exactly if a i = b i for all i = 1 , . . . , n (that is, if a = b ) or there exists and index k such that a i = b i for all i satisfying 1 ≤ i < k and a k ≺ k b k .

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