Skip to main content

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


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 \(\lambda _S\).

This is a preview of subscription content, access via your institution.


  1. Guo, Y. Q., Zhang D., and Shum, K. P., Some Studies on Disjunctive Degree and infix-i-Disjunctive Degrees of r-Disjunctive Languages (submitted).


  • Berstel J, Perrin D (1985) Theory of codes. Academic Press, Orlando

    MATH  Google Scholar 

  • Guo YQ, Shyr HJ, Thierrin G (1986) F-disjunctive languages. Int J Comput Math 18:219–237

    Article  Google Scholar 

  • Guo YQ, Reis CM, Thierrin G (1988) Relatively f-disjunctive languages. Semigroup Forum 37:289–299

    MathSciNet  Article  Google Scholar 

  • Howie JM (1991) Automata and languages. Clarendon Press, Oxford

    MATH  Google Scholar 

  • Ito M (1993) Dense and disjunctive properties of languages. In: Proceedings of the Fundamentals of Computation Theory, International Symposium, Fct ’93, Szeged, Hungary, August 23–27, 1993. DBLP 31–49

  • Jürgensen H, Yu SS (1990) Solid codes. J Inf Process Cybern 26(10):563–574

  • Lallement G (1979) Semigroups and combinatorial applications. Wiley, New York

    MATH  Google Scholar 

  • Liu Y, Guo YQ, Tsai YS (2007) Solid codes and the uniform density of fd-domains. Sci China Ser A 50(7):1026–1034

    MathSciNet  Article  Google Scholar 

  • Liu Y, Shum KP, Guo YQ (2008) Relatively regular languages and thin codes. Eur J Combin 29:261–267

    MathSciNet  Article  Google Scholar 

  • Reis CM (1987) A note on F-disjunctive languages. Semigroup Forum 36:159–165

    MathSciNet  Article  Google Scholar 

  • Reis CM (1990) F-disjunctive congruences and a generalization of monoids with length. Semigroup Forum 41:291–306

    MathSciNet  Article  Google Scholar 

  • Shyr HJ, Thierrin G (1977) Disjunctive languages and codes, fundamentals of computation theory. In: Proceeding of the 1977 Inter. FCT-conference, Poznan, Poland, lecture notes in computer science, No. 56. Springer, Berlin, pp 171–176

  • Shyr HJ, Yu SS (1990) Solid codes and disjunctive domains. Semigroup Forum 41(1):23–37

    MathSciNet  Article  Google Scholar 

  • Zhang D, Guo YQ, Shum KP (2014) On some decompositions of r-disjunctive languages. Bull Malays Math Sci Soc 37(3):727–746

    MathSciNet  MATH  Google Scholar 

  • Zhang D, Guo YQ, Shum KP (2017) Some results in r-disjunctive languages and related topics. Soft Comput 21(10):2477–2483

    Article  Google Scholar 

Download references


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).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Yuqi Guo.

Ethics declarations

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.

Additional information

Communicated by A. Di Nola.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the National Natural Science Foundation of China (Grant No. 11861071).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, Z., Guo, Y. & Leng, J. The applications of solid codes to r-R and r-D languages. Soft Comput 23, 10709–10716 (2019).

Download citation

  • Published:

  • Issue Date:

  • DOI:


  • Solid code
  • Principal congruence
  • Relatively regular language
  • Relatively disjunctive language