Partial orders
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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 .

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Presentation Transcript

Definitions
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).


Definitions1
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
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
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 diagrams1
Hasse Diagrams

  • What n yields a poset [Dn; | ] whose Hasse diagram is a cube of dimension:

    • 0

    • 1

    • 2

    • 3


Examples1
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
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
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
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
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
Reflexive-Transitive Closure

  • R0 = { (x, x) | x  S }

  • The reflexive-transitive closure, denoted R*, is: R* = R0  R+. That is,


Example
Example

R

a

b

c

d

e

R0

Self-loops not shown

a

b

c

d

e

R+

a

b

c

d

e

R*

Self-loops not shown

a

b

c

d

e


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