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.

Partial vs. Total Orderings
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.

If it is the case that $$\preceq$$ is a connected relation, i.e. that every pair of elements is related by $$\preceq$$, then $$\preceq$$ is called a total ordering.

If it is not the case that $$\preceq$$ is connected, then $$\preceq$$ is called a partial ordering.