The property of commutativity for some generalizations of BCK algebras

  • Andrzej WalendziakEmail author


We consider thirty generalizations of BCK algebras (RM, RML, BCH, BCC, BZ, BCI algebras and many others). We investigate the property of commutativity for these algebras. We also give 10 examples of proper commutative finite algebras. Moreover, we review some natural classes of commutative RML algebras and prove that they are equationally definable.


RM, RML, BCH, BCC, BZ, BCI algebra Commutativity Quasi-BCK* algebra Equational class 


Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.

Human and animal rights

This article does not contain any studies with human or animal participants performed by the author.


  1. Buşneag D, Rudeanu S (2010) A glimpse of deductive systems in algebra. Cent Eur J Math 8:688–705MathSciNetzbMATHGoogle Scholar
  2. Cīrulis J (2008) Implications in sectionally pseudocomplemented posets. Acta Sci Math (Szeged) 74:477–491MathSciNetzbMATHGoogle Scholar
  3. Cīrulis J (2010) Residuation subreducts of pocrigs. Bull Sect Logic 39:11–16MathSciNetzbMATHGoogle Scholar
  4. Dudek WA, Karamdin B, Bhatti SA (2011) Branches and ideals of weak BCC-algebras. Algebra Colloq 18(Spec 1):899–914MathSciNetCrossRefzbMATHGoogle Scholar
  5. Hu QP, Li X (1983) On BCH-algebras. Math Semin Notes 11:313–320MathSciNetzbMATHGoogle Scholar
  6. Imai Y, Iséki K (1966) On axiom system of propositional calculi. Proc Jpn Acad 42:19–22MathSciNetCrossRefzbMATHGoogle Scholar
  7. Iorgulescu A (2016a) New generalizations of BCI, BCK and Hilbert algebras—part I. J Mult Valued Logic Soft Comput 27:353–406MathSciNetzbMATHGoogle Scholar
  8. Iorgulescu A (2016b) New generalizations of BCI, BCK and Hilbert algebras—part II. J Mult Valued Logic Soft Comput 27:407–456MathSciNetzbMATHGoogle Scholar
  9. Iséki K (1966) An algebra related with a propositional culculus. Proc Jpn Acad 42:26–29CrossRefzbMATHGoogle Scholar
  10. Iséki K (1980) On BCI-algebras. Math Semin Notes 8:125–130zbMATHGoogle Scholar
  11. Jun YB, Roh EH, Kim HS (1998) On BH-algebras. Sci Math Jpn 1:347–354MathSciNetzbMATHGoogle Scholar
  12. Kim HS, Kim YH (2006) On BE-algebras. Sci Math Jpn e-2006:1299–1302Google Scholar
  13. Komori Y (1984) The class of BCC-algebras is not a variety. Math Jpn 29:391–394MathSciNetzbMATHGoogle Scholar
  14. Meng BL (2009) CI-algebras. Sci Math Jpn e–2009:695–701Google Scholar
  15. Meng BL (2010) Closed filters in CI-algebras. Sci Math Jpn e–2010:265–270MathSciNetzbMATHGoogle Scholar
  16. Meng J, Jun YB (1994) BCK algebras. Kyung Moon SA, SeoulzbMATHGoogle Scholar
  17. Piekart B, Walendziak A (2011) On filters and upper sets in CI-algebras. Algebra Discrete Math 11:97–103MathSciNetzbMATHGoogle Scholar
  18. Tanaka S (1975) A new class of algebras. Math Semin Notes 3:37–43MathSciNetGoogle Scholar
  19. Thomys J, Zhang X (2013) On weak-BCC-algebras. Sci World J 2013:10Google Scholar
  20. Walendziak A (2009) On commutative BE-algebras. Sci Math Jpn 69:281–284MathSciNetzbMATHGoogle Scholar
  21. Walendziak A (2018) Deductive systems and congruences in RM algebras. J Mult Valued Logic Soft Comput 30:521–539MathSciNetzbMATHGoogle Scholar
  22. Ye R (1991) Selected paper on BCI/BCK-algebras and computer logics. Shaghai Jiaotong University Press, Shaghai, pp 25–27Google Scholar
  23. Yutani H (1977) On a system of axioms of commutative BCK-algebras. Math Semin Notes 5:255–256MathSciNetzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Institute of Mathematics and Physics, Faculty of SciencesSiedlce University of Natural Sciences and HumanitiesSiedlcePoland

Personalised recommendations