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Daniel Moskovich
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A group $G$ is residually finite if, for any two elements $g$ and $g'$$g^\prime$ in $G$, there is a finite group $G'$$G^\prime$ and a (group) homomorphism $f: G \rightarrow G'$$f: G \rightarrow G^\prime$ such that $f(g)$ doesn't equal $f(g')$$f(g^\prime)$. The definition for a semigroup is analagous: just make $G$ and $G'$$G^\prime$ semigroups and make f$f$ 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'$$G^\prime$ and a group homomorphism from $G$ to $G'$$G^\prime$ 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?

Thanks!

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 $f: G \rightarrow G'$ 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?

Thanks!

A group $G$ is residually finite if, for any two elements $g$ and $g^\prime$ in $G$, there is a finite group $G^\prime$ and a (group) homomorphism $f: G \rightarrow G^\prime$ such that $f(g)$ doesn't equal $f(g^\prime)$. The definition for a semigroup is analagous: just make $G$ and $G^\prime$ 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 $G^\prime$ and a group homomorphism from $G$ to $G^\prime$ 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?

Thanks!

A group G$G$ is residually finite if, for any two elements g$g$ and g'$g'$ in G$G$, there is a finite group G'$G'$ and a (group) homomorphism f: G -> G'$f: G \rightarrow G'$ such that f(g)$f(g)$ doesn't equal f(g')$f(g')$. The definition for a semigroup is analagous: just make G$G$ and G'$G'$ 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$G$ which is not residually finite as a group but is residually finite as a semigroup (in other words there is a finite semigroup S$S$ and a semigroup homomorphism from G$G$ to S$S$ which separates elements, but there is no finite group G'$G'$ and a group homomorphism from G$G$ to G'$G'$ which separates elements)?

If S$S$ is a residually finite semigroup and G$G$ is a subgroup of S$S$, then G$G$ is residually finite as a semigroup. Is G$G$ residually finite as a group?

Thanks!

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?

Thanks!

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 $f: G \rightarrow G'$ 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?

Thanks!

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Charles Matthews
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residual Residual finiteness of groups vsversus residual finiteness of semigroups

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dave
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