The applications of solid codes to r-R and r-D languages

Abstract

A language S on a free monoid \(A^*\) is called a solid code if S is an infix code and overlap-free. A congruence \(\rho \) on \(A^*\) is called principal if there exists \(L\subseteq A^*\) such that \(\rho =P_L\), where \(P_L\) is the syntactic congruence determined by L. For any solid code S over A, Reis defined a congruence \(\sigma _S\) on \(A^*\) by means of S and showed it is principal (Semigroup Forum 41:291–306, 1990). A new simple proof of the fact that \(\sigma _S\) is principal is given in this paper. Moreover, two congruences \(\rho _S\) and \(\lambda _S\) on \(A^*\) defined by solid code S are introduced and proved to be principal. For every class of the classification of \({{\mathbf {D}}}_{\mathbf{r}}\) and \({{\mathbf {R}}}_{\mathbf{r}}\), languages are given by means of three principal congruences \(\sigma _S\), \(\rho _S\) and \(\lambda _S\).