# Definition:Strict Ordering

## Definition

### Definition 1

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

Then $\RR$ is a **strict ordering (on $S$)** if and only if $\RR$ satisfies the strict ordering axioms:

\((1)\) | $:$ | Asymmetry | \(\ds \forall a, b \in S:\) | \(\ds a \mathrel \RR b \) | \(\ds \implies \) | \(\ds \neg \paren {b \mathrel \RR a} \) | |||

\((2)\) | $:$ | Transitivity | \(\ds \forall a, b, c \in S:\) | \(\ds \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c} \) | \(\ds \implies \) | \(\ds a \mathrel \RR c \) |

### Definition 2

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

Then $\RR$ is a **strict ordering (on $S$)** if and only if $\RR$ satisfies the strict ordering axioms:

\((1)\) | $:$ | Antireflexivity | \(\ds \forall a \in S:\) | \(\ds \neg \paren {a \mathrel \RR a} \) | |||||

\((2)\) | $:$ | Transitivity | \(\ds \forall a, b, c \in S:\) | \(\ds \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c} \implies a \mathrel \RR c \) |

## Notation

Symbols used to denote a general strict ordering are usually variants on $\prec$, $<$ and so on.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, to denote a general strict ordering it is recommended to use $\prec$.

To denote the conventional strict ordering in the context of numbers, the symbol $<$ is to be used.

The symbol $\subset$ is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$, as it is ambiguous in the literature, and this can be a cause of confusion and conflict.

Hence the symbols $\subsetneq$ and $\subsetneqq$ are used for the (proper) subset relation.

\(\ds a\) | \(\prec\) | \(\ds b\) | can be read as: | \(\quad\) $a$ (strictly) precedes $b$ | ||||||||||

\(\ds a\) | \(\prec\) | \(\ds b\) | can also be read as: | \(\quad\) $b$ (strictly) succeeds $a$ |

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

### Notation for Inverse Strict Ordering

To denote the dual of an strict ordering, the conventional technique is to reverse the symbol.

Thus:

- $\succ$ denotes $\prec^{-1}$

and so:

- $a \prec b \iff b \succ a$

Similarly for the standard symbol used to denote a strict ordering on numbers:

- $>$ denotes $<^{-1}$

and so on.

## Strict vs. Weak Ordering

Some sources define an **ordering** as we on $\mathsf{Pr} \infty \mathsf{fWiki}$ define a **strict ordering**.

Hence, in contrast with such a **strict ordering**, the term **weak ordering** is often used in this context to mean what we define on $\mathsf{Pr} \infty \mathsf{fWiki}$ as an **ordering**.

It is essential to be aware of the precise definitions used by whatever text is being studied so as not to fall into confusion.

## Partial vs. Total Strict Ordering

It is not demanded of a strict ordering $\prec$, defined in its most general form on a set $S$, that *every* pair of elements of $S$ is related by $\prec$.

They may be, or they may not be, depending on the specific nature of both $S$ and $\prec$.

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

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

Beware that some sources use the word **partial** for a strict 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:

**Strict ordering**: a strict ordering whose nature (total or partial) is not specified

**Strict partial ordering**: a strict ordering which is specifically**not**total

**Strict total ordering**: a strict ordering which is specifically**not**partial.

## Also known as

Some sources call this an **antireflexive (partial) ordering**.

## Also see

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