2.9 Partial Orders 2.9 Partial Orders

By replacing symmetry by antisymmetry in the definition of an equiv- alence relation, we get the notion of a partial order. 2.9 Partial Orders

By replacing symmetry by antisymmetry in the definition of an equiv- alence relation, we get the notion of a partial order.

So a partial order ≤ on a set X is a binary relation on X such that: 2.9 Partial Orders

By replacing symmetry by antisymmetry in the definition of an equiv- alence relation, we get the notion of a partial order.

So a partial order . on a set X is a binary relation on X such that:

• a . a for all a ∈ X (reflexive) 2.9 Partial Orders

By replacing symmetry by antisymmetry in the definition of an equiv- alence relation, we get the notion of a partial order.

So a partial order . on a set X is a binary relation on X such that:

• a . a for all a ∈ X (reflexive)

• for all a, b ∈ X, if a . b and b . a then a = b (antisymmetric) 2.9 Partial Orders

By replacing symmetry by antisymmetry in the definition of an equiv- alence relation, we get the notion of a partial order.

So a partial order . on a set X is a binary relation on X such that:

• a . a for all a ∈ X (reflexive)

• for all a, b ∈ X, if a . b and b . a then a = b (antisymmetric)

• for all a, b, c ∈ X, if a . b and b . c then a . c (transitive). Examples. Examples.

• Any set of real numbers X equipped with the usual ≤. Examples.

• Any set of real numbers X equipped with the usual ≤.

In fact this order is total meaning that for all x, y ∈ X, either x ≤ y or y ≤ x (or possibly both!). Examples.

• Any set of real numbers X equipped with the usual ≤.

In fact this order is total meaning that for all x, y ∈ X, either x ≤ y or y ≤ x (or possibly both!).

• Let X = 2Y where Y is a set. This means X is all subsets of Y . Examples.

• Any set of real numbers X equipped with the usual ≤.

In fact this order is total meaning that for all x, y ∈ X, either x ≤ y or y ≤ x (or possibly both!).

• Let X = 2Y where Y is a set. This means X is all subsets of Y .

Then the inclusion relation ⊆ given by S ⊆ T if S is a subset of T is a partial order on X. Examples.

• Any set of real numbers X equipped with the usual ≤.

In fact this order is total meaning that for all x, y ∈ X, either x ≤ y or y ≤ x (or possibly both!).

• Let X = 2Y where Y is a set. This means X is all subsets of Y .

Then the inclusion relation ⊆ given by S ⊆ T if S is a subset of T is a partial order on X.

This partial order is not total: e.g. {1, 2} and {2, 3} are subsets of Y = {1, 2, 3} but neither contains the other. • The relation of divisibility on the natural numbers N is a partial order: • The relation of divisibility on the natural numbers N is a partial order:

m|n if m is a factor of n, i.e. if n = km for some k ∈ N. • The relation of divisibility on the natural numbers N is a partial order:

m|n if m is a factor of n, i.e. if n = km for some k ∈ N.

Again this partial order is not total • The relation of divisibility on the natural numbers N is a partial order:

m|n if m is a factor of n, i.e. if n = km for some k ∈ N.

Again this partial order is not total

(2 does not divide 3 and 3 does not divide 2). A set X together with a partial order . on it is called a partially or- dered set (X, .), or a poset. A set X together with a partial order . on it is called a partially or- dered set (X, .), or a poset.

If X is finite, there is a digraph associated with the poset (X, .). A set X together with a partial order . on it is called a partially or- dered set (X, .), or a poset.

If X is finite, there is a digraph associated with the poset (X, .).

But digraphs of partial orders are usually drawn in a different way: A set X together with a partial order . on it is called a partially or- dered set (X, .), or a poset.

If X is finite, there is a digraph associated with the poset (X, .).

But digraphs of partial orders are usually drawn in a different way: using a Hasse diagram. A set X together with a partial order . on it is called a partially or- dered set (X, .), or a poset.

If X is finite, there is a digraph associated with the poset (X, .).

But digraphs of partial orders are usually drawn in a different way: using a Hasse diagram.

For these, only put an edge from a to b if a 6= b, a . b, and there is nothing in between a and b, A set X together with a partial order . on it is called a partially or- dered set (X, .), or a poset.

If X is finite, there is a digraph associated with the poset (X, .).

But digraphs of partial orders are usually drawn in a different way: using a Hasse diagram.

For these, only put an edge from a to b if a 6= b, a . b, and there is nothing in between a and b, and then draw b higher than a, and omit the arrow. A set X together with a partial order . on it is called a partially or- dered set (X, .), or a poset.

If X is finite, there is a digraph associated with the poset (X, .).

But digraphs of partial orders are usually drawn in a different way: using a Hasse diagram.

For these, only put an edge from a to b if a 6= b, a . b, and there is nothing in between a and b, and then draw b higher than a, and omit the arrow.

The Hasse diagram contains complete information about .. For example, let X = {1, 2, 3, 4} with the partial order the usual ordering of real numbers: For example, let X = {1, 2, 3, 4} with the partial order the usual ordering of real numbers:

1 ≤ 2 ≤ 3 ≤ 4. For example, let X = {1, 2, 3, 4} with the partial order the usual ordering of real numbers:

1 ≤ 2 ≤ 3 ≤ 4.

The digraph would look like: For example, let X = {1, 2, 3, 4} with the partial order the usual ordering of real numbers:

1 ≤ 2 ≤ 3 ≤ 4.

The digraph would look like:

1 • • 2

4 • • 3 but the Hasse diagram is as follows: but the Hasse diagram is as follows:

• 4

• 3

• 2

• 1 Another example: X = {1, 2, 3, 4, 5, 6}, ordered by divisibility: Another example: X = {1, 2, 3, 4, 5, 6}, ordered by divisibility:

1|n for all n ∈ X, 2|2, 4, 6, 3|3, 6, 4|4, 5|5, 6|6. Another example: X = {1, 2, 3, 4, 5, 6}, ordered by divisibility:

1|n for all n ∈ X, 2|2, 4, 6, 3|3, 6, 4|4, 5|5, 6|6.

The Hasse diagram is as follows: Another example: X = {1, 2, 3, 4, 5, 6}, ordered by divisibility:

1|n for all n ∈ X, 2|2, 4, 6, 3|3, 6, 4|4, 5|5, 6|6.

The Hasse diagram is as follows:

4 6 • •

5 • 2 • • 3

• 1 A final example: A final example:

X = 2{1,2} = {∅, {1}, {2}, {1, 2}}, A final example:

X = 2{1,2} = {∅, {1}, {2}, {1, 2}}, ordered by inclusion. A final example:

X = 2{1,2} = {∅, {1}, {2}, {1, 2}}, ordered by inclusion.

Hasse diagram: A final example:

X = 2{1,2} = {∅, {1}, {2}, {1, 2}}, ordered by inclusion.

Hasse diagram:

{1, 2}

{1} {2}

{0} Equivalence relations and partial orders are examples of preorders: Equivalence relations and partial orders are examples of preorders: these satisfy only reflexivity and transitivity in general. Equivalence relations and partial orders are examples of preorders: these satisfy only reflexivity and transitivity in general.

So R∗ for any binary relation R on X is a preorder of X. Equivalence relations and partial orders are examples of preorders: these satisfy only reflexivity and transitivity in general.

So R∗ for any binary relation R on X is a preorder of X.

Preorders can be thought of as “partial orders on the equivalence classes of a set”. Equivalence relations and partial orders are examples of preorders: these satisfy only reflexivity and transitivity in general.

So R∗ for any binary relation R on X is a preorder of X.

Preorders can be thought of as “partial orders on the equivalence classes of a set”.

An example is R defined on all people by: Equivalence relations and partial orders are examples of preorders: these satisfy only reflexivity and transitivity in general.

So R∗ for any binary relation R on X is a preorder of X.

Preorders can be thought of as “partial orders on the equivalence classes of a set”.

An example is R defined on all people by: xRy if and only if x is at least as tall as y. Equivalence relations and partial orders are examples of preorders: these satisfy only reflexivity and transitivity in general.

So R∗ for any binary relation R on X is a preorder of X.

Preorders can be thought of as “partial orders on the equivalence classes of a set”.

An example is R defined on all people by: xRy if and only if x is at least as tall as y. (Neither symmetric nor antisymmetric.)