Partial Orders — Basics

Edward A. Lee UC Berkeley — EECS EECS 219D — Concurrent Models of Computation Last updated: January 23, 2014 Outline

Sets Join (Least Upper Bound) Relations and Functions Meet (Greatest Lower Bound) Notation Example of Join and Meet Directed Sets, Bottom Partial Order What is Order? Complete Partial Order Strict Partial Order Complete Partial Order Chains and Total Orders Alternative Definition Quiz Example Partial Orders — Basics

Sets

Frequently used sets:

• B = {0, 1}, the set of binary digits.

• T = {false, true}, the set of truth values.

• N = {0, 1, 2, ···}, the set of natural numbers.

• Z = {· · · , −1, 0, 1, 2, ···}, the set of integers.

• R, the set of real numbers.

• R+, the set of non-negative real numbers.

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Relations and Functions

• A binary relation from A to B is a subset of A × B.

• A partial function f from A to B is a relation where (a, b) ∈ f and (a, b0) ∈ f =⇒ b = b0. Such a partial function is written f : A*B.

• A total function or just function f from A to B is a partial function where for all a ∈ A, there is a b ∈ B such that (a, b) ∈ f.

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Notation • A binary relation: R ⊆ A × B.

• Infix notation: (a, b) ∈ R is written aRb.

• A symbol for a relation:

• ≤⊂ N × N • (a, b) ∈≤ is written a ≤ b.

• A function is written f : A → B, and the A is called its domain and the B its codomain. Rather than writing (a, b) ∈ f, we can equivalently write f(a) = b.

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Partial Order

A partial order on a set A is a relation from A to A satisfying the following properties. For all a, b, c ∈ A, the relation is

1. reflexive: a ≤ a

2. antisymmetric: a ≤ b and b ≤ a implies that a = b.

3. transitive: a ≤ b and b ≤ c implies that a ≤ c.

A partially ordered set or poset is (A, ≤), or (A, ≤A).

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What is Order? • 0 < 1 • 1 < ∞ • child < parent • child > parent • 11,000/3,501 is a better approximation to p than 22/7 • integer n is a divisor of integer m. • Set A is a subset of set B. • I know more about x than about y.

Which of these are partial orders?

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Strict Partial Order

A poset (A, ≤) induces another relation < called the strict partial order relation: ∀ a, a0 ∈ A, a < a0 ⇔ a ≤ a0 and a 6= a0 . (A, <) is called a strict poset.

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Chains and Total Orders

1. If a, a0 ∈ A satisfy either a ≤ a0 or a0 ≤ a, then a and a0 are comparable. Otherwise, they are incomparable.

2. A chain C ⊆ A is a subset of a poset (A, ≤) where any two members of the subset are comparable.

3.A total order is a poset (A, ≤) where A itself is a chain.

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Quiz

1. Is the set of integers with the usual numerical ordering a well-ordered set? (A well-ordered set is a set where every non-empty subset has a least element.)

2. Given a set A and its powerset (set of all subsets) ℘(A), is (℘(A), ⊆) a poset? A chain?

3. For A = {a, b, c} (an alphabet), find a well-ordered subset of (℘(A), ⊆).

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Quiz 1. Is the set of integers with the usual numerical ordering a well-ordered set? No. The set itself is a chain with no least element.

2. Given a set A and its powerset (set of all subsets) ℘(A), is (℘(A), ⊆) a poset? A chain? It is a poset, but not a chain.

3. For A = {a, b, c} (an alphabet), find a well-ordered subset of (℘(A), ⊆). One possibility: {∅, {a}, {a, b}, {a, b, c}}.

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Join (Least Upper Bound)

• Given a poset (A, ≤) and a subset B ⊆ A, an upper bound of B, if it exists, is an element a ∈ A such that for all b ∈ B, b ≤ a.

• A least upper bound or LUB, if it exists, is an upper bound a such that for all other upper bounds a0 we have a ≤ a0.

• If a set B ⊆ A has a least upper bound in the poset (A, ≤), then it is said to be joinable in (A, ≤).

• The LUB is called the join of B and written W B.

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Meet (Greatest Lower Bound)

• Given a poset (A, ≤) and a subset B ⊆ A, a lower bound of B, if it exists, is an element a ∈ A such that for all b ∈ B, a ≤ b.

• A greatest lower bound or GLB, if it exists, is a lower bound a such that for all other lower bounds a0 we have a0 ≤ a.

• If a set B ⊆ A has a GLB in the poset (A, ≤), the GLB is called the meet of B, written V B.

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Example of Join and Meet

Given a set A and a poset (℘(A), ⊆), then for any B ⊆ ℘(A),

• W B = ∪B (the union of subsets)

• V B = ∩B (the intersection of subsets)

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Directed Sets, Bottom

• A nonempty subset D ⊆ A of poset (A, ≤) is a directed set if every pair of elements in D has an upper bound in D.

• Equivalently, D is directed if every non-empty finite subset of D is joinable in D.

V • A pointed poset has a bottom element, often written ⊥A = A ∈ A, or simply ⊥.

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Complete Partial Order

A complete partial order or CPO (A, ≤) is a pointed poset where every directed subset is joinable in A.

Examples:

• Every finite pointed poset is a CPO.

• (N, ≤) is not a CPO.

• (N ∪ {∞}, ≤) is a CPO.

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Complete Partial Order Alternative Definition

A complete partial order or CPO (A, ≤) is a pointed poset where every chain has a LUB in A.

The equivalence of this definition is not trivial. See Davey and Priestly (2002), theorem 8.11.

The equivalence is trivial for some posets, where every directed set is a chain, such a prefix orders, which we will look at later.

Quiz: Is a pointed poset A where every chain is finite always a CPO?

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Complete Partial Order Alternative Definition

A complete partial order or CPO (A, ≤) is a pointed poset where every chain has a LUB in A.

The equivalence of this definition is not trivial. See Davey and Priestly (2002), theorem 8.11.

The equivalence is trivial for some posets, where every directed set is a chain, such a prefix orders, which we will look at later.

Quiz: Is a pointed poset A where every chain is finite always a CPO? Yes

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Example

Hasse diagram of a poset (A, ≤) where

A = {⊥, a, b, c, d, e, f, g, h}

Quiz: Is this a CPO?

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Example

Hasse diagram of a poset (A, ≤) where

A = {⊥, a, b, c, d, e, f, g, h}

Quiz: Is this a CPO? Yes. Pointed and finite.

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Example

Quiz: Is A well ordered?

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Example

Quiz: Is A well ordered? No. E.g., D = {e, f} has no least element.

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Example

Quiz: Does every pair of elements of A have a GLB?

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Example

Quiz: Does every pair of elements of A have a GLB? Yes. Such a poset is a lower semilattice.

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Example

Quiz: Does every pair of elements of A have a LUB?

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Example

Quiz: Does every pair of elements of A have a GLB? No. Such a poset would be an upper semilattice. If it is both a lower and upper semilattice, then it is a lattice.

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Example

Quiz: Is D = {e, f} joinable in A?

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Example

Quiz: Is D = {e, f} joinable in A? Yes. W D = a.

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Example

Quiz: Is D = {e, f} joinable in D?

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Example

Quiz: Is D = {e, f} joinable in D? No. No upper bound in D, so no least upper bound.

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References

Davey, B. A. and H. A. Priestly, 2002: Introduction to Lattices and Order. Cambridge University Press, 2nd ed.

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