Operation on Set for which Every Equivalence Relation is Congruence

Theorem

Let $S$ be a set with at least $3$ elements.

Let $\circ$ be an operation on $S$ such that every equivalence relation on $S$ is a congruence relation for $\circ$.

Then $\circ$ is one of the following:

the right operation $\to$

the left operation $\gets$

the constant operation $\sqbrk c$ for some $c \in S$.

Proof

First we note that from:

Equivalence Relation is Congruence for Constant Operation

Equivalence Relation is Congruence for Left Operation

Equivalence Relation is Congruence for Right Operation

every equivalence relation on $S$ is a congruence relation for the right operation, the left operation and the constant operation.

Let $\circ$ be an operation on $S$ which is not one of those three.

The validity of the material on this page is questionable. In particular: It is not a given that these three conditions hold at the same time. It also seems overly strong for the conclusion. But an immediate fix eludes me You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Questionable}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Then there exists $x_1, x_2, y_1, y_2 \in S$ such that:

$x_1 \circ y_1 = z_1$

$x_2 \circ y_2 = z_2$

where:

$z_1 \ne z_2$

$\map \lnot {z_1 = x_1 \land z_2 = x_2}$

$\map \lnot {z_1 = y_1 \land z_2 = y_2}$

The above are possible only if there are more than $3$ elements in $S$.

Let $\RR$ be an equivalence relation on $S$ such that:

$x_1 \mathrel \RR x_2$

$y_1 \mathrel \RR y_2$

but that:

$\map \lnot {z_1 \mathrel \RR z_2}$

Then $\RR$ is not a congruence relation.

The result follows from the Rule of Transposition.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.14$