Definition:Strict Ordering

From ProofWiki
Jump to: navigation, search

Definition

Definition 1

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:

\((1)\)   $:$   Asymmetry      \(\displaystyle \forall a, b \in S:\)    \(\displaystyle a \mathrel {\mathcal R} b \)   \(\displaystyle \implies \)   \(\displaystyle \neg \paren {b \mathrel {\mathcal R} a} \)             
\((2)\)   $:$   Transitivity      \(\displaystyle \forall a, b, c \in S:\)    \(\displaystyle \paren {a \mathrel {\mathcal R} b} \land \paren {b \mathrel {\mathcal R} c} \)   \(\displaystyle \implies \)   \(\displaystyle a \mathrel {\mathcal R} c \)             


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.


Also see