Monomorphisms of Boolean Algebras

Question

Suppose $f: B->C$ is a homomorphism between Boolean Algebras such that $ker(f)=0$. Is $f$ necessarily a monomorphism? Thank You.

Answer 1

If $f(a)=f(b)$, with $a\neq b$, then $a\nleq b$ or $b\nleq a$.
Suppose, without loss of generality, that $a \nleq b$. Thus $a \wedge b' \neq 0$.
Now, taking $c = a \wedge b'$, we have $$f(c) = f(a) \wedge f(b') = f(a) \wedge (f(b))' = f(a) \wedge (f(a))'=0.$$