Definition:Semigroup with respect to Equivalence Relation

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Let $C$ be a class.

Let $\thickapprox$ be an equivalence relation on $C$.

Let $\left({C, \cdot}\right)$ be a large magma.

Then $\left({C, \cdot}\right)$ is a semigroup with respect to $\thickapprox$ iff:

$\forall x, y,z \in C: \left( x \cdot y \right)\cdot z \thickapprox x \cdot \left(y \cdot z\right)$

Also see

Stronger properties