Abstract
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.
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References
Buşneag D, Rudeanu S (2010) A glimpse of deductive systems in algebra. Cent Eur J Math 8:688–705
Cīrulis J (2008) Implications in sectionally pseudocomplemented posets. Acta Sci Math (Szeged) 74:477–491
Cīrulis J (2010) Residuation subreducts of pocrigs. Bull Sect Logic 39:11–16
Dudek WA, Karamdin B, Bhatti SA (2011) Branches and ideals of weak BCC-algebras. Algebra Colloq 18(Spec 1):899–914
Hu QP, Li X (1983) On BCH-algebras. Math Semin Notes 11:313–320
Imai Y, Iséki K (1966) On axiom system of propositional calculi. Proc Jpn Acad 42:19–22
Iorgulescu A (2016a) New generalizations of BCI, BCK and Hilbert algebras—part I. J Mult Valued Logic Soft Comput 27:353–406
Iorgulescu A (2016b) New generalizations of BCI, BCK and Hilbert algebras—part II. J Mult Valued Logic Soft Comput 27:407–456
Iséki K (1966) An algebra related with a propositional culculus. Proc Jpn Acad 42:26–29
Iséki K (1980) On BCI-algebras. Math Semin Notes 8:125–130
Jun YB, Roh EH, Kim HS (1998) On BH-algebras. Sci Math Jpn 1:347–354
Kim HS, Kim YH (2006) On BE-algebras. Sci Math Jpn e-2006:1299–1302
Komori Y (1984) The class of BCC-algebras is not a variety. Math Jpn 29:391–394
Meng BL (2009) CI-algebras. Sci Math Jpn e–2009:695–701
Meng BL (2010) Closed filters in CI-algebras. Sci Math Jpn e–2010:265–270
Meng J, Jun YB (1994) BCK algebras. Kyung Moon SA, Seoul
Piekart B, Walendziak A (2011) On filters and upper sets in CI-algebras. Algebra Discrete Math 11:97–103
Tanaka S (1975) A new class of algebras. Math Semin Notes 3:37–43
Thomys J, Zhang X (2013) On weak-BCC-algebras. Sci World J 2013:10
Walendziak A (2009) On commutative BE-algebras. Sci Math Jpn 69:281–284
Walendziak A (2018) Deductive systems and congruences in RM algebras. J Mult Valued Logic Soft Comput 30:521–539
Ye R (1991) Selected paper on BCI/BCK-algebras and computer logics. Shaghai Jiaotong University Press, Shaghai, pp 25–27
Yutani H (1977) On a system of axioms of commutative BCK-algebras. Math Semin Notes 5:255–256
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Walendziak, A. The property of commutativity for some generalizations of BCK algebras. Soft Comput 23, 7505–7511 (2019). https://doi.org/10.1007/s00500-018-03691-9
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DOI: https://doi.org/10.1007/s00500-018-03691-9