Monadic pseudo BCI-algebras and corresponding logics

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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

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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.

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

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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

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