# Definition:Commutative Semigroup with respect to Equivalence Relation

## Definition

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 commutative semigroup with respect to $\thickapprox$ iff:

it is a semigroup with respect to $\thickapprox$
$\forall x, y \in C: x \cdot y \thickapprox y \cdot x$