Definition:Semigroup with respect to Equivalence Relation
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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 $(2)$