Definition:Strict Ordering

Definition
Let $$S$$ be a set.

An strict ordering on $$S$$ is a relation $$\mathcal{R}$$ on $$S$$ such that:


 * $$\mathcal{R}$$ is irreflexive, that is, $$\forall a \in S: \neg a \mathcal{R} a$$
 * $$\mathcal{R}$$ is transitive, that is, $$\forall a, b, c \in S: a \mathcal{R} b \and b \mathcal{R} c \implies a \mathcal{R} c$$
 * $$\mathcal{R}$$ is asymmetric, that is, $$\forall a, b \in S: a \mathcal{R} b \implies \neg b \mathcal{R} a$$

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


 * "$$a \prec b$$" can be read as: "$$a$$ precedes $$b$$".

Alternatively, "$$a \prec b$$" can be read as: "$$b$$ succeeds $$a$$".

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


 * $$a \prec b \iff b \succ a$$

If, for two elements $$a, b \in S$$, $$\neg a \prec b$$, then the symbols $$a \not \prec b$$ and $$b \not \succ a$$ can be used.

Partial vs. Total Orderings
Note that this definition of "strict ordering" does not demand that every pair of elements of $$S$$ is related by $$\prec$$. The way we have defined a strict ordering, they may be, or they may not be, depending on the context.

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

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