A bijective monoid homomorphism is called a monoid isomorphism. Any bijective ring homomorphism is a ring isomorphism. A "K"-algebra isomorphism is a bijective "K"-algebra homomorphism. A semigroup homomorphism between monoids preserves identity if it is a monoid homomorphism. Conversely, the kernel of any semigroup homomorphism is a semigroup congruence. The restriction to this stabilizer of above group homomorphism is an isomorphism. Condition 3 is equivalent to the fact that this bundle homomorphism is an isomorphism. Let formula 34 be a homomorphism of commutative ring. Equivalently, it is a diffeomorphism which is also a group homomorphism. In fact, it is the kernel of a surjective group homomorphism from S to S. The kernel of a homomorphism is always a congruence. There is a natural rng homomorphism defined by . Moreover, they are closed under union, intersection, concatenation, substitution, inverse homomorphism, and Kleene plus. This induces a homomorphism on the modular group . This justifies the name "embedding" for an arbitrary homomorphism of fields.