8 6 Partial Orderings Definition Partial ordering a

8. 6 Partial Orderings

Definition Partial 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 a. Rb • Also a≺b iff a≺ b and a≠b

Examples and non-examples of posets (S, ≼) • 1. (Z, ≤) • 2. (Z, ≥) proof

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 1. (Z, ≤ ) using the usual ≤ 2. (Z+, |) 3. (P(S), ) where S={1, 2, 3} incomparable (if any):

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<discrete Discreet<discreetness Discrete<discretion

Lexicographic order • Suppose (A 1, ≼ 1) and (A 2, ≼ 2) are two posets. • Let (a 1, a 2), (b 1, b 2) � A 1 x. A 2 • Let (a 1, a 2) ≺ (b 1, b 2) in case either a 1 ≺ 1 b 1 or (a 1=b 1 and a 2 ≺ 2 b 2) • Letter or number examples

(A 1 x. A 2, ≼) 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 diagram 4 | 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 u� S and a ≼ u for all a� A, 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 | f | e | c g | d | b a • find lub and glb upper bounds of A: lub(A)= lower bounds of A: glb(A)= upper bounds of B lower bounds of B

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+ 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 a. Rb. 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 •

Advising example •