Definition:Strict Ordering/Antireflexive and Transitive

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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.


Also see


Sources