Definition:Preordering

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Definition

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


Definition 1

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

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


Definition 2

$\mathcal R$ is a preordering on $S$ if and only if:

$(1): \quad \mathcal R \circ \mathcal R = \mathcal R$
$(2): \quad \Delta_S \subseteq \mathcal R$

where:

$\circ$ denotes relation composition
$\Delta_S$ denotes the diagonal relation on $S$.


Preordered Set

Let $S$ be a set.

Let $\precsim$ be a preordering on $S$.


Then the relational structure $\struct {S, \precsim}$ is called a preordered set.


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.


Examples

Finite Set Difference on Natural Numbers

Consider the relation $\RR$ on the powerset of the natural numbers:

$\forall a, b \in \powerset \N: a \mathrel \RR b \iff a \setminus b \text { is finite}$

where $\setminus$ denotes set difference.


Then $\RR$ is a preordering on $\powerset \N$, but not an ordering on $\powerset \N$.


Also see

  • Results about preorderings can be found here.