Definition:Semigroup with respect to Equivalence Relation

Definition

Let $C$ be a class.

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

Let $\struct {C, \cdot}$ be a large magma.

Then $\struct {C, \cdot}$ is a semigroup with respect to $\thickapprox$ if and only if:

$\forall x, y,z \in C: \paren {x \cdot y} \cdot z \thickapprox x \cdot \paren {y \cdot z}$

Also see

Stronger properties

• Definition:Semigroup
• Definition:Commutative Semigroup with respect to Equivalence Relation
• Definition:Commutative Semigroup

Sources

• 1981: Stanley Burris and H.P. Sankappanavar: A Course in Universal Algebra: $\text {II} \ \S 1$ Example $(2)$