# Definition:Total Ordering

(Redirected from Definition:Linear Ordering)

## Definition

Let $\mathcal R \subseteq S \times S$ 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:

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

## Also known as

Some sources call this 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.

## Also see

• Results about total orderings can be found here.