Ordering of infinite sets

Introduction

Ordered sets

Totally ordered sets

A totally ordered set is one where the relation is defined on all pairs:

\(\forall a \forall b (a\le b)\lor (b\le a)\)

Note that totality implies reflexivity.

Partially ordered sets (poset)

A partially ordered set, or poset, is one where the relation is defined between each element and itself.

\(\forall a (a\le a)\)

That is, every element is related to itself.

These are also called posets.

Well-ordering

A well-ordering on a set is a total order on the set where the set contains a minimum number. For example the relation \(\le \) on the natural numbers is a well-ordering because \(0\) is the minimum.

The relation \(\le \) on the integers however is not a well-ordering, as there is no minimum number in the set.

Intervals

For a totally ordered set we can define a subset as being all elements with a relationship to a number. For example:

\([a,b]=\{x:a\le x \land x\le b\}\)

This denotes a closed interval. Using the definition above we can also define an open interval:

\((a,b)=\{x:a< x \land x< b\}\)

Infinitum and supremum

Infinitum

Consider a subset \(S\) of a partially ordered set \(T\).

The infinitum of \(S\) is the greatest element in \(T\) that is less than or equal to all elements in \(S\).

For example:

\(\inf [0,1]=0\)

\(\inf (0,1)=0\)

Supremum

The supremum is the opposite: the smallest element in \(T\) which is greater than or equal to all elements in \(S\).

\(\sup [0,1]=1\)

\(\sup (0,1)=1\)

Max and min

If the infinitum of a set \(S\) is in \(S\), then the infinimum is the minimum of set \(S\). Otherwise, the minimum is not defined.

\(\min [0,1]=0\)

\(\min (0,1)\) isn’t defined.

Similarly:

\(\max [0,1]=1\)

\(\max (0,1)\) isn’t defined.