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On the groups of codes with empty kernel. (English) Zbl 1202.20071
A word $$v\in A^*$$ is an internal factor of a word $$x\in A^*$$ iff $$x=uvw$$ for some nonempty words $$u,w$$. The kernel of a set $$X\subset A^*$$ is the set of words from $$X$$ which are internal factors of some word from $$X$$. It is shown, that if $$X$$ is a code with empty kernel, $$F$$ the set of internal factors of words from $$X$$ and $$\varphi$$ the syntactic morphism of the submonoid $$X^*$$, then any group $$G$$ contained in $$\varphi(A^*\setminus F)$$ is cyclic. A subclass of codes with empty kernel are semaphore codes, thus this is a generalization of a result of M. P. Schützenberger [Inf. Control 7, 23-26 (1964; Zbl 0122.15004)].

##### MSC:
 20M35 Semigroups in automata theory, linguistics, etc. 20M05 Free semigroups, generators and relations, word problems 68Q45 Formal languages and automata
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##### References:
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