Skip to main content

Monadic pseudo BCI-algebras and corresponding logics

Abstract

We introduce the notion of monadic pseudo BCI-algebras and study some related properties. Then, we introduce monadic filters and monadic congruences of monadic pseudo BCI-algebras and discuss the relations between them. We proved that there is a one-to-one correspondence between the set of closed m-congruence relations and the set of normal closed m-filters in a monadic pseudo BCI-algebra. Moreover, we introduce a notion of strong residuated mappings and study the relation between monadic operators and strong residuated mappings in pseudo BCI-algebras. Let A be a pseudo BCI-algebra and \(f:A\rightarrow A\) be a mapping, we obtain that \((f, f^+)\) is a monadic operator on A if and only if f is a strong residuated mapping on A where \(f^+\) is the residual of f. Also we exhibit an axiom system of monadic pseudo BCI-logic, which enrich the language of pseudo BCI-logics. Based on the monadic pseudo BCI-algebras, we prove the completeness and soundness of the monadic pseudo BCI-logic propositional system. Finally, using provable formula set, normal subset and monadic subset in the set of all formulas of a monadic pseudo BCI-logic \(\mathcal {L}\), we characterize filters, normal filters and monadic normal filters in a monadic pseudo BCI-algebra, respectively.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  • Blok WJ, Pigozzi D (1989) Algebraizable logics. Memoirs of the American Mathematical Society. American Mathematical Society, Providence

    MATH  Google Scholar 

  • Blyth TS (2005) Lattices and ordered algebraic structures. Springer, London

    MATH  Google Scholar 

  • Curry HB, Feys R, Craig W (1958) Combinatory logic, vol 1. North Holland, Amsterdam

    MATH  Google Scholar 

  • Dudek WA, Jun YB (2008) Pseudo BCI-algebras. East Asian Math J 24:187–190

    MATH  Google Scholar 

  • Dvurečenskij A, Vetterlein T (2002) Algebras in the positive cone of po-groups. Order 19:127–146

    MathSciNet  Article  MATH  Google Scholar 

  • Dymek G (2014) On compatible deductive systems of pseudo-BCI-algebras. J Mult-Valued Log Soft Compt 22:167–187

    MathSciNet  MATH  Google Scholar 

  • Dymek G, Kozanecka-Dymek A (2013) Pseudo-BCI-logic. Bull Sect Log 42:33–41

    MathSciNet  MATH  Google Scholar 

  • Georgescu G, Iorgulescu A (2001) Pseudo-BCKalgebras: an extension of BCKalgebras. In: Proceedings of DMTCS01: combinatorics, computability and logic, Springer, London, pp 97–114

  • Iorgulescu A (2008) Algebras of Logic as BCK-algebras. Academy of Economic Studies Bucharest, Editura

    MATH  Google Scholar 

  • Iséki K (1966) An algebra related with a propositional calculus. Proc Jpn Acad 42:26–29

    MathSciNet  Article  MATH  Google Scholar 

  • Iséki K (1980) On BCI-algebras. Math Sem Notes Kobe Univ 8(1):125–130

    MathSciNet  MATH  Google Scholar 

  • Kabziński JK (1983) BCI-algebras from the point of view of logic. Bull Sect Log Pol Acad Sci Inst Philos Soc 12:126–129

    MathSciNet  MATH  Google Scholar 

  • Meng J, Jun YB (1994) BCK-algebras. Kyungmoon Sa, Seoul

    MATH  Google Scholar 

  • Meredith CA, Prior AN (1963) Notes on the axiomatics of the propositional calculus. Notre Dame J Formal Log 4:171C187

    MathSciNet  Article  MATH  Google Scholar 

  • Prior AN (1962) Formal logic, 2nd edn. Clarendon Press, Oxford

    MATH  Google Scholar 

  • Xin XL, Li YJ (2017) States on pseudo-BCI algebras. Eur J Pure Appl Math 10(3):455–472

    MathSciNet  MATH  Google Scholar 

Download references

Funding

This research is partially supported by a Grant of the National Key Research and Development Program of China (Grant 2016YFB0800700), National Natural Science Foundation of China (11571281,61602359), the Fundamental Research Funds for the Central Universities (JB181503) and the 111 project (Grants B08038 and B16037).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xiaolong Xin.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Ethical approval

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

Informed consent

Informed consent was obtained from all individual participants included in the study.

Additional information

Publisher's Note

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

Communicated by A. Di Nola.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Xin, X., Fu, Y., Lai, Y. et al. Monadic pseudo BCI-algebras and corresponding logics. Soft Comput 23, 1499–1510 (2019). https://doi.org/10.1007/s00500-018-3189-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-018-3189-7

Keywords

  • pseudo BCI-algebra
  • Monadic operator
  • Monadic filter
  • Residuated mapping
  • pseudo BCI-logic