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 irreflexive, 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.

Also see

• Strict Ordering is Asymmetric

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 14$: Order
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 7$