Complement of Strict Total Ordering

Theorem

Let $\struct {S, \prec}$ be a relational structure such that $\prec$ is a strict total ordering.

We need to show that $\nprec$ is a weak total ordering.

First we check in turn each of the criteria for an ordering:

By Complement of Reflexive Relation, it follows directly that $\nprec$ is reflexive.

Let $a \nprec b$ and $b \nprec c$.

Now, suppose that $a \prec c$.

As $b \nprec c$, from the Trichotomy Law, one of the following is the case:

$b = c$ and so $a \prec b$;

$c \prec b$ and $a \prec b$ from transitivity of $\prec$.

Both are an immediate contradiction.

So $a \nprec c$ and so $\nprec$ is transitive.

Suppose $a \nprec b$.

Then from the Trichotomy Law, either $a = b$ or $b \prec a$.

Similarly, if $b \nprec a$ we have that either $a = b$ or $a \prec b$.

As $\prec$ is asymmetric, $b \prec a \iff \neg a \prec b$.

So if $a \nprec b$ and $b \nprec a$, it follows that $a = b$.

So $\nprec$ is antisymmetric.

The fact that $\nprec$ is connected follows directly from the Trichotomy Law.

$\blacksquare$