Question

Condition of semigroup homomorphism should be ____________

(a) f(x * x) = f(x * y)

(b) f(x) = f(y)

(c) f(x) * f(y) = f(y)

(d) f(x * y) = f(x) * f(y)

I had been asked this question in my homework.

I'm obligated to ask this question of Group Axioms in division Groups of Discrete Mathematics

Answer 1

The correct option is (d) f(x * y) = f(x) * f(y)

For explanation I would say: Consider two semigroups (S,∗) and (S’,∗’). A function f: S -> S’ is called a semigroup homomorphism if f(a∗b) = f(a)∗f(b). Suppose f is also one-to-one and onto. Then f is called an isomorphism between S and S’ and S and S’ are said to be isomorphic semigroups.