Definition:Preordering

Definition

Let $S$ be a set.

A preordering on $S$ is a relation $\mathcal R$ on $S$ such that:

• $\mathcal R$ is reflexive, that is, $\forall a \in S: a \mathcal R a$;
• $\mathcal R$ is transitive, that is, $\forall a, b, c \in S: a \mathcal R b \land b \mathcal R c \implies a \mathcal R c$.

Symbols used to define such a general preordering relation are often variants on $\lesssim$, $\precsim$ or $\precapprox$.

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

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

A preordered set is a set $S$ endowed with a preorder.

This can be expressed as the relational structure $\left({S, \precsim}\right)$.

Extensions

Ordering

If a preordering is also antisymmetric, that is, $\forall a, b \in S: a \mathcal R b \land b \mathcal R a \implies a = b$, then $\mathcal R$ is an ordering.

Equivalence Relation

If a preordering is also symmetric, that is, $\forall a, b \in S: a \mathcal R b \implies b \mathcal R a$, then $\mathcal R$ is an equivalence relation.

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.

Alternative names

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.

Steven A. Gaal: Point Set Topology (1964) 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.

Sources

• Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 7$: Exercise $7$