$\RR$ 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||\(\displaystyle \forall a, b, c \in S:\)||\(\displaystyle a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c \)|
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 $a \not \sim b$
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.
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.
- Results about preorderings can be found here.
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$: Exercise $7$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: Further exercises: $5$
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Definition $6$
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.7$