On codes having no finite completions (Q1239130)

Let \(X^*\) denote the free monoid generated by the set \(X\). A subset \(A\) of \(X^*\) is called a code if \(A^*\) is a free submonoid of \(X^*\) of base \(A\). A code \(A\) in \(X^*\) is called complete if it is no proper subset of any other code in \(X\). The main result of this paper is that there exist finite codes which are not contained in any finite complete code or equivalently that in a free monoid there exist finitely generated free submonoids that are not contained in any maximal finitely generated free submonoid. Also a characterization of all finite complete codes \(A\) on the alpha-bet \(X=\{x,y\}\) of the form \(A\subset x^* \cup x^*yx^*\) is given.