# Definition:Ordering/Definition 2

## Definition

Let $S$ be a set.

An **ordering on $S$** is a relation $\mathcal R$ on $S$ such that:

- $(1): \quad \mathcal R \circ \mathcal R = \mathcal R$
- $(2): \quad \mathcal R \cap \mathcal R^{-1} = \Delta_S$

where:

- $\circ$ denotes relation composition
- $\mathcal R^{-1}$ denotes the inverse of $\mathcal R$
- $\Delta_S$ denotes the diagonal relation on $S$.

## Notation

Symbols used to denote a general ordering relation are usually variants on $\preceq$, $\le$ and so on.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, to denote a general ordering relation it is recommended to use $\preceq$ and its variants:

- $\preccurlyeq$
- $\curlyeqprec$

To denote the conventional ordering relation in the context of numbers, the symbol $\le$ is to be used, or its variants:

- $\leqslant$
- $\leqq$
- $\eqslantless$

The symbol $\subseteq$ is universally reserved for the subset relation.

- $a \preceq b$

can be read as:

**$a$ precedes, or is the same as, $b$**.

Similarly:

- $a \preceq b$

can be read as:

**$b$ succeeds, or is the same as, $a$**.

If, for two elements $a, b \in S$, it is not the case that $a \preceq b$, then the symbols $a \npreceq b$ and $b \nsucceq a$ can be used.

## Partial vs. Total Ordering

It is not demanded of an ordering $\preceq$, defined in its most general form on a set $S$, that *every* pair of elements of $S$ is related by $\preceq$. They may be, or they may not be, depending on the specific nature of both $S$ and $\preceq$.

If it *is* the case that $\preceq$ is a connected relation, that is, that every pair of distinct elements is related by $\preceq$, then $\preceq$ is called a total ordering.

If it is *not* the case that $\preceq$ is connected, then $\preceq$ is called a partial ordering.

Beware that some sources use the word **partial** for an ordering which **may or may not** be connected, while others insist on reserving the word **partial** for one which is specifically **not** connected.

It is wise to be certain of what is meant.

As a consequence, on $\mathsf{Pr} \infty \mathsf{fWiki}$ we resolve any ambiguity by reserving the terms for the objects in question as follows:

**Partial ordering**: an ordering which is specifically**not**total

**Total ordering**: an ordering which is specifically**not**partial.

## Also see

- Results about
**orderings**can be found here.