Trivial Ordering

Theorem

The trivial ordering is an ordering $\mathcal R$ in a poset $\left({S, \mathcal R}\right)$ such that:

$\forall a, b \in S: a \mathcal R b \iff a = b$

That is, there is no ordering defined on any two distinct elements of the set $S$.

The trivial ordering is universally compatible.

Proof

To prove that the trivial ordering is in fact an ordering, we need to checking each of the criteria for an ordering:

Reflexivity

$\forall a \in S: a \mathcal R a$:

From its definition, we have $\forall a, b \in S: a = b \implies a \mathcal R b$.

Thus, as $a = a$, we have $\forall a \in S: a \mathcal R a$.

So reflexivity is proved.

Transitivity

$\forall a, b, c \in S: a \mathcal R b \land b \mathcal R c \implies a \mathcal R c$:

From the definition:

• $a \mathcal R b \iff a = b$
• $b \mathcal R c \iff b = c$

So as $a = b \land b = c \implies a = c$ from transitivity of equals, we have that $a \mathcal R c$ and thus transitivity is proved.

Antisymmetry

$\forall a, b \in S: a \mathcal R b \land b \mathcal R a \implies a = b$:

From the definition:

• $a \mathcal R b \iff a = b$.
• $b \mathcal R a \iff b = a$.

Antisymmetry follows from symmetry of equals.

• The trivial ordering is by definition the same as the diagonal relation, and is therefore universally compatible.

$\blacksquare$

Sources

• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 3.1$: Example $1$