8.6 Partial Orderings

DefinitionPartial ordering– a relation R on a set S that is Reflexive,

Antisymmetric, and Transitive

Examples?• R={(a,b)| a is a subset of b }

• R={(a,b)| a divides b } on {1,2,3,4}– R={(1,1),(1,2),(1,3),(1,4),(2,2),…}

• R={(a,b)| a≤ b }

• R={(a,b)| a=b+1 }

Partially ordered set (poset)

• (S,R) -- a set S and a relation R on S, that is R, A, and T.

• Often we use (S, ≼) • Note: is a generic symbol for R≼• It includes the usual ≤, but it is more general. It also

covers other poset relations: divides, subset,…

• We say a b iff aRb≼• Also a b iff a≺ ≺ b and a≠b

Examples and non-examples of posets (S, ≼)

• 1. (Z, ≤) proof

• 2. (Z, ≥)

More examples

• 3. (Z, |) where | is “divides”

• 4. ( Z+ , |)

…examples• 5. (P(S), ) where S={1,2,3} and P(S) is the

power set

• 6. (P(S), ) where S is a set and P(S) is the power set

Comparable

• Def: The elements a and b of a poset (S, ≼) are said to be “comparable” if either a ≼b or b ≼a.• Otherwise, they are “incomparable.”

Comparable, incomparable elements• For each set, find

comparable elements incomparable (if any):

1. (Z, ≤ ) using the usual ≤ 2. (Z+, |)

3. (P(S), ) where S={1,2,3}

totally (linearly) ordered set

• Def:• A poset (S, ≼) is a totally (linearly) ordered set

if every two elements of S are comparable. • ≼ is then a total order, and S is a chain.

Are these examples total orders or not?

• (Z, ≤ )

• (Z+, |)

Lexicographic Order (dictionary)

Things to consider:Longer lengths or different lengths in words Ex: Discreet<discreteDiscreet<discreetnessDiscrete<discretion

Lexicographic order

• Suppose (A1, ≼1) and (A2, ≼2) are two posets.

• Let (a1, a2), (b1, b2) A1xA2

• Let (a1, a2) ≺ (b1, b2) in case either a1 ≺ 1 b1 or (a1=b1 and a2 ≺ 2 b2)

• Letter or number examples

(A1xA2, ≼) is a poset

• Proof Method?• Proof – see book

Hasse diagram

• Hasse diagram—a diagram that contains sufficient information to find a partial ordering

• Algorithm:– create a digraph with directed edges pointing up– remove all loops (reflexive is assumed)– remove any (a,c) where (a,b) and (b,c) are present

(transitivity assumed)– remove arrows (direction up is assumed)

Ex. 1. S={1,2,3,4}; poset (S, ≤)

Original digraph reduced diagram4|

3|2|1

Ex. 2: (S, ≼) where S={1,2,3,4,6,8,12} and ≼ ={(a,b)|a divides b}

Shorthand: ({1,2,3,4,6,8,12}, | ) 8 12| |4 6| |2 3|1

Ex 3: Hasse diagram of (P({a,b,c}), )

•

Ex. 4: Hasse of ({2,4,5,10,12,20,25,}, | )

•

Maximal, minimal…

• Def:• Let (S, ≼) be a poset and a S.– a is maximal in (S, ≼) if there does not exist b S such that a ≺ b.– a is minimal in (S, ≼) if there does not exist b S such that b ≺ a.– a is the greatest element of (S, ≼) if b ≼ a for all b S.– a is the least element of (S, ≼) if a ≼ b for all b S.

• • Find examples of maximal, greatest elements,… in

above examples.

greatest element

• Claim: The greatest element, when it exists, is unique.

• Proof:– Method?

• Similarly, the least element, when it exists, is unique.

Upper bound,…

• Def: Let (S, ≼) be a poset and A S.– If uS and a ≼ u for all aA,u is an upper bound of A.– If l S and l ≼ a for all a A, l is an lower bound of A.– x is a least upper bound of A , lub(A), if x is an upper

bound and x ≼ z for every upper bound z of A.– y is a greatest lower bound of A , glb(A), if y is a lower

bound and z ≼ y for every lower bound z of A.

– Remark: lub and glb are unique when they exist.

Ex. 5(S, ≼ )A={b,d,g}, B=(d,e}

h i upper bounds of A:| lub(A)=

g f lower bounds of A:| | glb(A)=d e| | upper bounds of Bb c

lower bounds of Ba

• find lub and glb

Ex. 6: A={4,6,8} with “divides” relation

lub(A)=glb(A)=

Note: lub=?glb=?

Well-ordered set

Def: (S, ≼) is well-ordered set if it is a poset such that ≼ is a total ordering and every nonempty subset of S has a least element.

Find Ex and non-ex.:• (Z+, ≤)• (Z, ≤)• (Z+ x Z+, lexicographic order)• (R+, ≤)

Topological sorting

Use: for project ordering Def:A total ordering ≼ is compatible with the partial order R

if a ≼ b whenever aRb.The construction of such a total order is called a

topological sorting. Lemma: Every finite non-empty poset (S, ≼ ) has a

minimal element.

({2,4,5,10,12,20,25}, | )Recall Hasse diagram for ({2,4,5,10,12,20,25}, | )

Create several topological sorts.

House Ex- book

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Advising example

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