Partial Orders

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# Partial Orders Definitions A relation R on set A is a partial order if ... - PowerPoint PPT Presentation

Partial Orders. Definitions. A relation R on set A is a partial order if it is: reflexive antisymmetric transitive. A is called a partially ordered set or poset . [A;R] means A is partially ordered by R. Example: [ P({1,2});  ] ( draw its digraph ). Definitions .

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### Partial Orders

Definitions
• A relation R on set A is a partial order if it is:
• reflexive
• antisymmetric
• transitive.
• A is called a partially ordered set or poset.
• [A;R] means A is partially ordered by R.
• Example: [ P({1,2});  ] (draw its digraph).
Definitions ...
• is the prototype of a partial order
• a, bA are comparable under  if either ab or ba.
• Otherwise, they are said to be incomparable
• If a, bA are comparable, then [A; ] is totally ordered and A is a chain.
Examples
• [ Z;  ],
• where Z is the set of integers
•  has the usual meaning.
• Z is totally ordered.
• [ Z+; | ],
• where Z+ is a set of positive integers
• | is the (evenly) divides operator (e.g., 2 | 10 )
Hasse Diagrams
• A Hasse diagram is a directed graph (digraph), where
• self-loops are omitted
• arcs implied by transitivity are omitted.
• Let Dn denote the set of positive divisors of n.
• Draw Hasse diagram for [ D8; | ]
• Draw Hasse diagram for [ D6; | ]
Hasse Diagrams
• What n yields a poset [Dn; | ] whose Hasse diagram is a cube of dimension:
• 0
• 1
• 2
• 3
Examples
• Let S be a set. [ P(S);  ] is a poset.
• Draw the Hasse diagram for:
• S = 
• S = {1}
• S = {1, 2}
• S = {1, 2, 3}
• What do you think the Hasse diagram looks like for S = {1, 2, 3, 4}? For S = {1, . . ., n}?
Computer Science Example
• Consider this sequence of Java assignments:

a = b + d + c;

d = a*(b + d) + c;

e = (b + d)*c;

(Draw an operator graph for these statements.)

• Sequence these operations in a way that is compatible with their partial order.
• Where are the longest paths?
Topological Sorting
• Let G = (V,E) be a directed graph where
• v V represents a task;
• (u, v)  E means that task u must be completed before v can be started.
• G cannot have cycles.
• Problem: Find a schedule for G that respects the partial order.
Composing Relations
• Let R be a relation from A to B.
• Let S be a relation from B to C.
• The composition of R & S, denoted RS, is

RS = { xRSy | zB, xRz  zSy }.

• Example:
• R = { (1,2), (3,4), (2,4) }
• S = { (2,4), (2,3), (4,1) }
• RS = { (1,4), (1,3), (3,1), (2,1) }
The Transitive Closure
• Let R be a relation from S to S.
• RR is usually denoted R2.
• Ri+1 = RiR.
• The transitive closure, R+, of a binary relation R is:
Reflexive-Transitive Closure
• R0 = { (x, x) | x  S }
• The reflexive-transitive closure, denoted R*, is: R* = R0  R+. That is,
Example

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R+

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R*

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