Download 6.1 Partially Ordered Sets

 A relation R on a set A is called a partial order if R is
Reflexive, Antisymmetric and Transitive.
 Reflexive – has a cycle of 1 at every vertex.
Example: (1,1), (2,2)
 Antisymmetric – contains no back arrows – all one way
paths and can contain cycles of 1.
Example: (1,1), (1,2), (2,3)
 Transitive – if you have (a,b) and (b,c), you must have (a,c).
Example:
a,b b,c a,c
(1,2), (2,3), (1,3)
 The set A together with the partial order R is called a
partially ordered set or poset. This is denoted (A,R) or
(A, ≤).
 The relation of divisibility (aRb if and only if a divides
b) is a partial order on Z+ .
A = {1,2,3,4} x ≤ y iff x|y
B = {1,7,11,14} x ≤ y in the regular way
x
y
x
y
a,b
a,b
a,b
a,b
(1,7) ≤ (3,14)
(2,7) ≤ (3,14)
Because 1|3 and 7 ≤ 14.
Because 2 does not divide 3.
 a and b of set A are said to be comparable if a ≤ b or b ≤
a. Every pair of elements does not need to be
comparable aRb iff a|b is a partial order on Z+ .
 2 and 7 are not comparable since 2|7 and 7|2.
 Partial means some elements may not be comparable.
Lexicographic (lex sick o graphic)
Means alphabetic order.
park < part
help < hind
Jump < mump
Hasse Diagrams
 Hasse diagrams, for simplicity, remove one cycles,
remove transitive property aRc, remove arrows, and all
edges point upwards.
 Jt does not mean one cycles, transitive property and
arrows don’t exist….they still exist.
A = {1,2,3,4,12}
a and b are ∈ of A
a ≤ b if and only if a|b
Draw the Hasse diagram of the poset(A, ≤).
(1,1),(1,2),(1,3),(1,4),(1,12),
(2,2),(2,4),(2,12),
(3,3),(3,12),
(4,4),(4,12),
(12,12)
 Cross off one cycles and (a,c) of transitive property.
(1,1),(1,2),(1,3),(1,4),(1,12),
(2,2),(2,4),(2,12),
(3,3),(3,12),
(4,4),(4,12),
(12,12)
12
12
4
4
3
2
3
1
2
1
Finite linear ordered set
1 2 3
1
2
3
(1,1), (1,2), (1,3),
(2,2), (2,3)
(3,3)
3
3
2
2
1
1
 The process of constructing a linear order is called
topological sorting. With topological sorting, you
need to be able to insert a relation in the linear order
without breaking an existing relation.
3
 1 is related to 2
2
 1 is also related to 3
1
3
 4 could be placed after 1 if 4 is related to 1 and this
would not break the relations of 1 to 2 and 1 to 3,
meaning 4 is related to 2 and 3
2
4
1
Isomorphism
 A = {1,2,3,6}
≅
f
Under divisibility, x ≤ y iff x|y
 B = {{}, {x},{y}, {x,y}}
Under subset, (a ≤ b) iff a ≤ b
 Hasse diagrams of A & B
1
2
3
6
≅
{}
{x}
{y}
{x,y}
{x,y}
6
3
2
1
{y}
{x}
{}
The function A B is one to one. The function f is an isomorphism.
Incomparable
a ≤ b and b ≤ a. There is no relation between a and b so
we cannot compare them.
b
a
c
d
 A poset (A, R -1) is called a dual poset…..it is the
inverse.
f
b
a
e
d
e
d
f
a
b
c
c