Poset&lattice

Poset&lattice - Poset and Lattice Lecture 7...

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Poset and Lattice Lecture 7 Discrete Mathematical Structures
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¤˜– „“
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Partial Order Reflexive, antisymmetric and transitive Generalization of “less than or equal to” Denotation: Example 1: set containment Note: not any two of sets are “comparable” Example 2: divisibility on Z +
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Partially Ordered Set a p artially o rdered set (poset) is a set with a partial order defined on it. Denotation: (A, ) Examples (Z, ) or (Z, ) (Z + , | )
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Dual poset Let R be a partial order on a set A R -1 is also a partial order Reflexive, anti-symmetric, transitive Dual of poset (A,R): (A, R -1 ) Dual of the partial order of R: R -1
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Product Partial Order Given two posets, ( A , A ) and ( B , B ), we can define a new partial order on A × B : ( a , b ) ( a ’, b ’) iff. a A a ’ in A and b B b ’ in B It is easy to prove that ( A × B, ) is a poset Lexicographic order, as simplified: Given a partial order on a alphabet A , then is a simplified “dictionary” order: (a,b) (a’,b’) iff. a a’ and a≠a’ or a=a’ and Denoted as a a’
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How about the order of Let S = {a,b,c,d,……,z}, linearly order: a ≤ b, b ≤c, ……, y ≤z Let S 4 = S×S×S×S – “part” : (p,a,r,t) S 4 ; “park” : (p,a,r,k) S 4 Lexicographic order applied park part So as to: S n or S* parking parting park parking
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Hasse Diagrams Partial order can be represented by common relation diagram Examples: A = {a,b,c}, partial order: {(a,a),(b,b),(c,c),(a,b),(b,c),(a,c)}
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Hasse Diagrams a b c Simplify: Omit all cycle with length 1 Omit all edges implied by the transitive property (path with length >1) Omit all arrows with default up and down direction a b c
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9 12 8 10 11 3 6 2 4 5 1 7 Divisibility on {1,2,3,. ..12}
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comparability If (A, ) is a poset, the elements a and b are comparable if a b or b a Lineal order: if every pairs of elements of A is comparable. chain
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Containment on ρ ({a,b,c}) {a,b,c} {a,b} {a,c} {b,c} {a} {b} {c} φ
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Divisibility on positive divisors of 30 5 30 6 15 10 3 2 1 D 30
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{a,b,c } {a,b} {a,c} {b,c} {a} {b} {c} φ 5 30 6 15 10 3 2 1 D 30
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Isomorphism Let ( A , ) and ( A’, ’) be posets and let f : A A ’ be a one-to-one correspondence between A and A ’. The function f is called an isomorphism from ( A , ) to ( A’, ’) if for any a and b in A, a b iff. f ( a ) ’ f ( b ). The two posets are called isomorphic posets.
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Isomorphism Z + and the set of positive even number are isomorphic under “≤” Define f:Z + → {even number}, f(a)=2a f is an one-one correspondence relation f is onto relation f is one to one : f(a) = f(b) implies a = b for any a,b Z + , a≤b iff f(a) ≤f(b) So, we get it
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Examples 5 30 6 15 10 3 2 1 D 30 {a,b,c } {a,b} {a,c} {b,c} {a} {b} {c} φ
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Poset&lattice - Poset and Lattice Lecture 7...

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