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As pointed out by Mark Sapir in his answerhis answer to a related question, every residually finite divisible semigroup is idempotent (hence uniquely divisible). On another hand, it is not difficult to prove that any idempotent Abelian semigroup is residually finite. So it is natural to ask whether or not the same holds even in the case where the semigroup operation is not commutative. Does it happen to be a well-established result? Is there a trivial counterexample that I can't see?

As pointed out by Mark Sapir in his answer to a related question, every residually finite divisible semigroup is idempotent (hence uniquely divisible). On another hand, it is not difficult to prove that any idempotent Abelian semigroup is residually finite. So it is natural to ask whether or not the same holds even in the case where the semigroup operation is not commutative. Does it happen to be a well-established result? Is there a trivial counterexample that I can't see?

As pointed out by Mark Sapir in his answer to a related question, every residually finite divisible semigroup is idempotent (hence uniquely divisible). On another hand, it is not difficult to prove that any idempotent Abelian semigroup is residually finite. So it is natural to ask whether or not the same holds even in the case where the semigroup operation is not commutative. Does it happen to be a well-established result? Is there a trivial counterexample that I can't see?

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Salvo Tringali
Salvo Tringali
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Idempotent semigroups: Are they all residually finite?

As pointed out by Mark Sapir in his answer to a related question, every residually finite divisible semigroup is idempotent (hence uniquely divisible). On another hand, it is not difficult to prove that any idempotent Abelian semigroup is residually finite. So it is natural to ask whether or not the same holds even in the case where the semigroup operation is not commutative. Does it happen to be a well-established result? Is there a trivial counterexample that I can't see?