Soft Computing https://doi.org/10.1007/s00500-018-3691-y FOUNDATIONS The property of commutativity for some generalizations of BCK algebras Andrzej Walendziak 1 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 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. Keywords RM, RML, BCH, BCC, BZ, BCI algebra · Commutativity · Quasi-BCK* algebra · Equational class 1 Introduction Imai and Iséki (1966) and Iséki (1966) introduced BCK and BCI algebras as algebras connected to certain kinds of logics. Several years later, Hu and Li (1983) defined BCH alge- bras. It is known that BCK and BCI algebras are contained in the class of BCH algebras. To solve some problems on BCK algebras, Komori (1984) introduced BCC algebras. These algebras (also called BIK + -algebras) are an algebraic model of BIK + -logic. Later on, Ye (1991) defined BZ alge- bras, which are a common generalization of BCI and BCC algebras. In the literature, BZ algebras are also called weak BCC algebras [cf. Dudek et al. (2011) or Thomys and Zhang (2013)]. Next, the new class of algebras called BE algebras was introduced by Kim and Kim (2006). As a generalization of BE algebras, Meng (2009) defined CI algebras. Bu¸ sneag and Rudeanu (2010) introduced pre-BCK algebras, which are particular cases of BE algebras. Recently, Iorgulescu (2016a, b) introduced new generalizations of BCI or of BCK algebras. All of the algebras mentioned above are contained in the class of RM algebras (a RM algebra is an algebra ( A;→, 1) of type (2, 0) satisfying the equations: x → x = 1 and 1 → x = x ). Deductive systems and congruences in Communicated by A. Di Nola. B Andrzej Walendziak walent@interia.pl 1 Institute of Mathematics and Physics, Faculty of Sciences, Siedlce University of Natural Sciences and Humanities, 3 Maja 54, 08110 Siedlce, Poland some subclasses of the class of RM algebras were studied by Walendziak (2018). Tanaka (1975) introduced the notion of commutativity in the theory of BCK algebras. Commutative BCI, BCC and BE algebras were considered in Iséki (1980), Meng and Jun (1994) and Walendziak (2009), respectively. In this paper, we investigate the property of commutativ- ity for various generalizations of BCK algebras. We present 10 examples of proper commutative finite algebras. More- over, we review some natural classes of commutative RML algebras and prove that they are equationally definable. 2 Preliminaries Let A = ( A;→, 1) be an algebra of type (2, 0). We consider the following list of properties (Iorgulescu 2016a) that can be satisfied by A: (An) x → y = 1 = y → x ⇒ x = y , (B) ( y → z ) →[(x → y ) → (x → z )]= 1, (BB) ( y → z ) →[(z → x ) → ( y → x )]= 1, (D) y → (( y → x ) → x ) = 1, (Ex) x → ( y → z ) = y → (x → z ), (K) x → ( y → x ) = 1, (L) x → 1 = 1, (M) 1 → x = x , (Re) x → x = 1, (*) y → z = 1 ⇒ (x → y ) → (x → z ) = 1, (**) y → z = 1 ⇒ (z → x ) → ( y → x ) = 1, (Tr) x → y = 1 = y → z ⇒ x → z = 1. 123