# 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 the following two conditions hold:

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

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

### Definition 2

Let $\mathcal R$ be a relation on a set $S$.

Then $\mathcal R$ is a **strict ordering (on $S$)** if and only if the following two conditions hold:

\((1)\) | $:$ | Antireflexivity | \(\displaystyle \forall a \in S:\) | \(\displaystyle \neg \paren {a \mathrel {\mathcal R} a} \) | ||||

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

## 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 universally reserved for the (proper) subset relation.

- $a \prec b$

can be read as:

**$a$ (strictly) precedes $b$**.

Similarly:

- $a \prec b$

can also be read as:

**$b$ (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 inverse 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.

## 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**.