Definition:Preordering

Definition

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

Definition 1

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

$(1)$	$:$	$\RR$ is reflexive	$\displaystyle \forall a \in S:$	$\displaystyle a \mathrel \RR a $
$(2)$	$:$	$\RR$ is transitive	$\displaystyle \forall a, b, c \in S:$	$\displaystyle 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 $\left({S, \precsim}\right)$ 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, i.e. that every pair of elements is related by $\precsim$, then $\precsim$ is called a total preordering.

If it is 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

Definition:Preorder Category, interpreting preorders as categories.

Results about preorderings can be found here.

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