A property of three-element codes (Q1083216)

Let w be a word and A a language of a finitely generated free monoid \(\Sigma^*\). We say that w is ambiguously covered by A if there exist words \(\alpha\) and \(\beta\) in A, with \(\alpha\neq \beta\), such that \(w\in pref(\alpha A^+)\cap pref(\beta A^+)\). We show that if A is a three- element code, then any two words which are ambiguously covered by A are comparable, i.e. one of them is a prefix of the other. This property is characteristic for three-element codes.