Residual finiteness of groups versus residual finiteness of semigroups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T17:56:17Z http://mathoverflow.net/feeds/question/34291 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34291/residual-finiteness-of-groups-versus-residual-finiteness-of-semigroups Residual finiteness of groups versus residual finiteness of semigroups dave 2010-08-02T18:21:36Z 2010-08-03T07:50:06Z <p>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:</p> <p>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)?</p> <p>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?</p> <p>Thanks!</p> http://mathoverflow.net/questions/34291/residual-finiteness-of-groups-versus-residual-finiteness-of-semigroups/34299#34299 Answer by Tsuyoshi Ito for Residual finiteness of groups versus residual finiteness of semigroups Tsuyoshi Ito 2010-08-02T20:26:18Z 2010-08-02T20:49:56Z <p>I first posted this as a comment, but I guess that this is an answer.</p> <p>If a group is residually finite as a semigroup, it is residually finite as a group. This is an immediate consequence of the following easy fact: if G is a group and φ:G→S is a semigroup homomorphism, then the image φ(G) is a group and φ is a group homomorphism from G to φ(G). I guess that the latter fact is in any textbook on semigroups, though I do not have one at hand.</p>