Definition:Total Ordering

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Definition

Let $\RR \subseteq S \times S$ be a relation on a set $S$.


Definition 1

$\RR$ is a total ordering on $S$ if and only if:

$(1): \quad \RR$ is an ordering on $S$
$(2): \quad \RR$ is connected

That is, $\RR$ is an ordering with no non-comparable pairs:

$\forall x, y \in S: x \mathop \RR y \lor y \mathop \RR x$


Definition 2

$\RR$ is a total ordering on $S$ if and only if:

\(\text {(1)}: \quad\) \(\ds \RR \circ \RR\) \(\subseteq\) \(\ds \RR\)
\(\text {(2)}: \quad\) \(\ds \RR \cap \RR^{-1}\) \(\subseteq\) \(\ds \Delta_S\)
\(\text {(3)}: \quad\) \(\ds \RR \cup \RR^{-1}\) \(=\) \(\ds S \times S\)


Class Theory

In the context of class theory, the definition follows the same lines:

Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be a relation.


Let $\RR$ be such that:

$(1): \quad \RR$ is an ordering on $\Field \RR$
$(2): \quad \forall x, y \in \Field \RR: x \mathop \RR y \lor y \mathop \RR x$ (that is, $x$ and $y$ are comparable)

where $\Field \RR$ denotes the field of $\RR$.


Then $\RR$ is a total ordering.


Also known as

Some sources refer to a total ordering as a linear ordering, or a simple ordering.


If it is necessary to emphasise that a total ordering $\preceq$ is not strict, then the term weak total ordering may be used.


Examples

Usual Ordering on Real Numbers

Let $\R$ denote the set of real numbers.

The usual ordering $\le$ on $\R$ and its dual $\ge$ are total orderings on $\R$.


Monarchy

Let $K$ denote the set of British monarchs.

Let $\MM$ denote the relation on $K$ defined as:

$a \mathrel \MM b$ if and only if $a$ was monarch after or at the same time as $b$.


Its dual $\MM^{-1}$ is defined as:

$a \mathrel {\MM^{-1} } b$ if and only if $a$ was monarch before or at the same time as $b$.


Then $\MM$ and $\MM^{-1}$ are total orderings on $K$.


Also see

  • Results about total orderings can be found here.