Case 1: The operator is a commutative semigroup.

Case 2: The operator is not a binary operation.

Case 3: The operator is not a semigroup.

Case 4: The operator is not a commutative semigroup.

Case 5: The operator is not a commutative semigroup.

Case 6: The operator is not a semigroup.

Case 7: The operator is a commutative semigroup.

Case 8: The operator is a commutative semigroup.

Case 9: The operator is not a binary operation.