A group G is residually finite if, for any two elements g and g' in G, there is a finite group G' and a (group) homomorphism f: G -> G' such that f(g) doesn't equal f(g'). The definition for a semigroup is analagous: just make G and G' semigroups and make f a semigroup homomorphism. I was wondering if there is a good reference which will answer questions like the following:
Is there a group G which is not residually finite as a group but is residually finite as a semigroup (in other words there is a finite semigroup S and a semigroup homomorphism from G to S which separates elements, but there is no finite group G' and a group homomorphism from G to G' which separates elements)?
If S is a residually finite semigroup and G is a subgroup of S, then G is residually finite as a semigroup. Is G residually finite as a group?