# Definition:Strict Ordering/Antireflexive and Transitive

## Definition

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:

 $(1)$ $:$ 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.