Diagonal Relation is Ordering and Equivalence

Theorem

Let $\struct {S, \Delta_S}$ be a relational structure where $\Delta_S$ is the diagonal relation, defined as:

$\forall x, y \in S: \paren {x, y} \in \Delta_S \iff x = y$

Then $\Delta_S$ is the only relation on $S$ which is both an equivalence and an ordering.

Proof

From Trivial Ordering is Universally Compatible and Diagonal Relation is Equivalence we know that the diagonal relation possesses these properties.

We now need to show that it is the only relation on $S$ which possesses these properties.

Both an equivalence and an ordering are reflexive and transitive.

Also:

An equivalence relation is symmetric

An ordering is antisymmetric.

The result follows from Relation is Symmetric and Antisymmetric iff Coreflexive.

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$: Exercise $1$