Definition:Preordering

Definition
Let $$S$$ be a set.

A preordering (or preorder, or quasi-ordering) 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 $$\precsim$$ or $$\precapprox$$.

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


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

Ordering
If a preordering is also antisymmetric, that is, $$\forall a, b \in S: a \mathcal{R} b \and b \mathcal{R} a \implies a = b$$, 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 Orderings
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.