Advertisement

Monadic pseudo BCI-algebras and corresponding logics

  • Xiaolong Xin
  • Yulong Fu
  • Yanyan Lai
  • Juntao Wang
Foundations
  • 42 Downloads

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.

Keywords

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

Notes

Acknowledgements

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).

Compliance with ethical standards

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.

References

  1. Blok WJ, Pigozzi D (1989) Algebraizable logics. Memoirs of the American Mathematical Society. American Mathematical Society, ProvidencezbMATHGoogle Scholar
  2. Blyth TS (2005) Lattices and ordered algebraic structures. Springer, LondonzbMATHGoogle Scholar
  3. Curry HB, Feys R, Craig W (1958) Combinatory logic, vol 1. North Holland, AmsterdamzbMATHGoogle Scholar
  4. Dudek WA, Jun YB (2008) Pseudo BCI-algebras. East Asian Math J 24:187–190zbMATHGoogle Scholar
  5. Dvurečenskij A, Vetterlein T (2002) Algebras in the positive cone of po-groups. Order 19:127–146MathSciNetCrossRefzbMATHGoogle Scholar
  6. Dymek G (2014) On compatible deductive systems of pseudo-BCI-algebras. J Mult-Valued Log Soft Compt 22:167–187MathSciNetzbMATHGoogle Scholar
  7. Dymek G, Kozanecka-Dymek A (2013) Pseudo-BCI-logic. Bull Sect Log 42:33–41MathSciNetzbMATHGoogle Scholar
  8. Georgescu G, Iorgulescu A (2001) Pseudo-BCKalgebras: an extension of BCKalgebras. In: Proceedings of DMTCS01: combinatorics, computability and logic, Springer, London, pp 97–114Google Scholar
  9. Iorgulescu A (2008) Algebras of Logic as BCK-algebras. Academy of Economic Studies Bucharest, EditurazbMATHGoogle Scholar
  10. Iséki K (1966) An algebra related with a propositional calculus. Proc Jpn Acad 42:26–29MathSciNetCrossRefzbMATHGoogle Scholar
  11. Iséki K (1980) On BCI-algebras. Math Sem Notes Kobe Univ 8(1):125–130MathSciNetzbMATHGoogle Scholar
  12. Kabziński JK (1983) BCI-algebras from the point of view of logic. Bull Sect Log Pol Acad Sci Inst Philos Soc 12:126–129MathSciNetzbMATHGoogle Scholar
  13. Meng J, Jun YB (1994) BCK-algebras. Kyungmoon Sa, SeoulzbMATHGoogle Scholar
  14. Meredith CA, Prior AN (1963) Notes on the axiomatics of the propositional calculus. Notre Dame J Formal Log 4:171C187MathSciNetCrossRefzbMATHGoogle Scholar
  15. Prior AN (1962) Formal logic, 2nd edn. Clarendon Press, OxfordzbMATHGoogle Scholar
  16. Xin XL, Li YJ (2017) States on pseudo-BCI algebras. Eur J Pure Appl Math 10(3):455–472MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Xiaolong Xin
    • 1
  • Yulong Fu
    • 2
  • Yanyan Lai
    • 1
  • Juntao Wang
    • 1
  1. 1.School of MathematicsNorthwest UniversityXi’anChina
  2. 2.School of Cyber EngineeringXidian UniversityXi’anChina

Personalised recommendations