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.
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.
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.
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:
\((a,b)=\{x:a< x \land x< b\}\)
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:
\(\inf [0,1]=0\)
\(\inf (0,1)=0\)
The supremum is the opposite: the smallest element in \(T\) which is greater than or equal to all elements in \(S\).
\(\sup [0,1]=1\)
\(\sup (0,1)=1\)
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.