# Operation on Set for which Every Equivalence Relation is Congruence

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## 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 meYou 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 $11.14$