Definition:Ordering

Let $$S$$ be a set.

An 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, $$\forall a, b, c \in S: a \mathcal{R} b \land b \mathcal{R} c \Longrightarrow a \mathcal{R} c$$
 * $$\mathcal{R}$$ is antisymmetric, that is, $$\forall a, b \in S: a \mathcal{R} b \land b \mathcal{R} a \Longrightarrow a = b$$

Symbols frequently used to define such a general ordering relation are variants on $$\preceq$$ or $$\le$$, although the latter is usually used in the context of numbers.

"$$a \preceq b$$" can be read as: "$$a$$ precedes, or is the same as, $$b$$".

Alternatively, "$$a \preceq b$$" can be read as: "$$b$$ succeeds, or is the same as, $$a$$".

A symbol for an ordering can be reversed, and the sense is likewise inverted:

$$a \preceq b \iff b \succeq a$$

If, for two elements $$a, b \in S$$, $$\lnot a \preceq b$$, then the symbols $$a \not \preceq b$$ and $$b \not \succeq a$$ can be used.

Note that this definition of "ordering" does not demand that every pair of elements of $$S$$ is related by $$\preceq$$. The way we have defined an ordering, they may be, or they may not be, depending on the context.

Some authors prefer to use the term "partial ordering" for what is defined here as an "ordering", but it can be argued that it is possible to allow more flexible thinking an attempt is made to decouple the actual ordering relation from the set it is defined on.