Diagonal Relation is Universally Compatible
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The validity of the material on this page is questionable.
It needs to be pointed out that the definition of universally compatible relation may not be correct as such. I will address it when I work up to that chapter in my book.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.
Theorem
The diagonal relation $\Delta_S$ on a set $S$ is universally compatible with every operation on $S$.
Proof
It needs to be pointed out that the definition of universally compatible relation may not be correct as such. I will address it when I work up to that chapter in my book.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.
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Let $\struct {S, \circ}$ be any algebraic structure.
| \(\displaystyle \) | \(\) | \(\displaystyle x_1 \Delta_S x_2 \land y_1 \Delta_S y_2\) | |||||||||||
| \(\displaystyle \) | \(\leadsto\) | \(\displaystyle x_1 = x_2 \land y_1 = y_2\) | Definition of Diagonal Relation | ||||||||||
| \(\displaystyle \) | \(\leadsto\) | \(\displaystyle x_1 \circ y_1 = x_2 \circ y_2\) | (consequence of equality) | ||||||||||
| \(\displaystyle \) | \(\leadsto\) | \(\displaystyle \paren {x_1 \circ y_1} \mathrel {\Delta_S} \paren {x_2 \circ y_2}\) | Definition of Diagonal Relation |
$\blacksquare$
