Definition:Strict Ordering

Definition
Let $S$ be a set.

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


 * $(1): \quad \mathcal R$ is antireflexive, that is, $\forall a \in S: \neg a \mathcal R a$
 * $(2): \quad \mathcal R$ is transitive, that is, $\forall a, b, c \in S: a \mathcal R b \land b \mathcal R c \implies a \mathcal R c$

Alternatively, $(1)$ may be replaced by:


 * $(1'): \quad \mathcal R$ is asymmetric, that is, $\forall a, b \in S: a \mathcal R b \implies \neg b \mathcal R a$

Equivalence of these definitions is proved in Equivalence of Strict Ordering Definitions.

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:
 * $a$ strictly precedes $b$

or:
 * $b$ strictly 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 $\prec$ is connected, then $\prec$ is called a strict partial ordering.

Beware that some sources use the word partial for an ordering which may or may not be connected, while others insist on reserving the word partial for one which is specifically not connected. It is wise to be certain of what is meant.

Alternative names
Some sources call this an antireflexive (partial) ordering.

Also see

 * Strict Ordering is Asymmetric