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Given a finite alphabet and a language
,
the centralizer of is defined as the maximal language commuting with it.
We prove that if the primitive root of the smallest word of (with respect to a lexicographic order) is prefix distinguishable in then the centralizer of
is as simple as possible, that is, the submonoid
.
This lets us obtain a simple proof of a known result concerning the centralizer of nonperiodic three-word languages.
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