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Topologies for the free monoid. (English) Zbl 0739.20032
Let $$L$$ be a recognizable subset of a free semigroup $$A^*$$, $$M$$ the syntactic monoid of $$L$$, $$E(M)$$ is the set of idempotents of $$M$$ and $$P$$ is the image of $$L$$ in $$M$$. It is conjectured that if for each $$s$$, $$t\in M$$ and each $$e\in E(M)$$, $$set\in P$$ implies $$st\in P$$, then $$L$$ is closed in $$\mathcal T$$, where the topology $$\mathcal T$$ on $$A^*$$ is defined by all the monoid morphisms from $$A^*$$ into a discrete finite group. The conjecture is proven in two particular cases: if $$P$$ is a submonoid of $$M$$, or if the idempotents of $$M$$ commute. The conjecture has several interesting consequences, for instance if it holds, then $$K(M)=D(M)$$ ( Rhodes’ Conjecture). Here $$K(M)=\cap 1\tau^{-1}$$, where the intersection is taken over all relational morphisms $$\tau$$ from $$M$$ into a group and $$D(M)$$ is the smallest submonoid of $$M$$ satisfying the condition: for every $$s$$, $$t\in M$$ such that either $$sts=s$$ or $$tst=t$$, from $$u\in D(M)$$ follows $$sut\in D(M)$$.
Reviewer: J.Henno (Naasa)

##### MSC:
 20M35 Semigroups in automata theory, linguistics, etc. 20M05 Free semigroups, generators and relations, word problems 20M15 Mappings of semigroups
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