On the simplest centralizer of a language
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.