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 $\lambda _S$.