Definition:Preordering/Definition 1

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Definition

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


$\RR$ is a preordering on $S$ if and only if $\RR$ satifies the preordering axioms:

\((1)\)   $:$   $\RR$ is reflexive      \(\ds \forall a \in S:\) \(\ds a \mathrel \RR a \)      
\((2)\)   $:$   $\RR$ is transitive      \(\ds \forall a, b, c \in S:\) \(\ds a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c \)      


Notation

Symbols used to denote a general preordering relation are usually variants on $\lesssim$, $\precsim$ or $\precapprox$.


A symbol for a preordering can be reversed, and the sense is likewise inverted:

$a \precsim b \iff b \succsim a$


The notation $a \sim b$ is defined as:

$a \sim b$ if and only if $a \precsim b$ and $b \precsim a$

The notation $a \prec b$ is defined as:

$a \prec b$ if and only if $a \precsim b$ and $a \not \sim b$


Partial vs. Total Preorderings

Note that this definition of preordering does not demand that every pair of elements of $S$ is related by $\precsim$.

The way we have defined a preordering, they may be, or they may not be, depending on the context.

If it is the case that $\precsim$ is a connected relation, that is, that every pair of elements is related by $\precsim$, then $\precsim$ is called a total preordering.

If it is specifically not the case that $\precsim$ is connected, then $\precsim$ is called a partial preordering.


Also known as

A preordering is also known as a preorder.

Either name can be seen with a hyphen: pre-ordering and pre-order.

Some sources use the term quasiorder or quasi-order.


1964: Steven A. Gaal: Point Set Topology uses the term reflexive partial ordering, but as this can so easily be confused with the concept of a partial ordering this term is not recommended.


Also see

  • Results about preorderings can be found here.


Sources

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