Soft Computing

, Volume 23, Issue 21, pp 10709–10716 | Cite as

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

  • Zuhua Liu
  • Yuqi GuoEmail author
  • Jing Leng


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 \(\lambda _S\).


Solid code Principal congruence Relatively regular language Relatively disjunctive language 



The authors thank the referees for their very careful and in-depth recommendations. This work was supported by the National Natural Science Foundation of China (Grant No. 11861071).

Compliance with ethical standards

Conflict of interest

Author Zuhua Liu declares that he has no conflict of interest. Author Yuqi Guo declares that he has no conflict of interest. Author Jing Leng declares that she has no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouChina
  2. 2.Department of MathematicsKunming UniversityKunmingChina

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