RM ALGEBRAS AND COMMUTATIVE MOONS

Some generalizations of BCI algebras (the RM, BH, CI, BCH, BH**, BCH**, and *aRM** algebras) satisfying the identity (x → 1) → y = (y → 1) → x are considered. The connections of these algebras and various generalizations of commutative groups (such as, for example, involutive commutative moons and commutative (weakly) goops) are described. In particular, it is proved that an RM algebra verifying this identity is equivalent to an involutive commutative moon. Mathematics Subject Classification (2020): 03G25, 06A06, 06F35

In 1983, Q. P. Hu and X. Li [2] defined BCH algebras, which are a generalization of BCI algebras. Later on, in 1998, Y. B. Jun et al. [7] introduced the notion of BH algebras. It is known that BCI and BCH algebras are contained in the class of BH algebras. Next, in 2009, the new class of algebras called CI algebras was introduced by B. L. Meng [11]. These algebras are a generalization of BCH algebras (hence also a generalization of BCI algebras). BH and CI algebras are also called aRM and RME algebras, respectively (see [3], [4], [14]). All of the algebras mentioned above are contained in the class of RM algebras (an RM algebra is an algebra (A; →, 1) of type (2, 0) satisfying the identities: x → x = 1 and 1 → x = x).
Deductive systems and congruences in RM algebras were studied by A. Walendziak [12]. The implicative and commutative properties for some subclasses of the class of RM algebras were investigated in [13,14]. In 1985, T. Lei and C. Xi [9] defined psemisimple BCI algebras and proved that p-semisimple BCI algebras are equivalent with commutative groups. The p-semisimple BCI algebras have been extensively investigated in many papers (for example [1], [8], [10], [16]).
In this paper, we consider some generalizations of BCI algebras (the RM, BH, CI, BCH, BH**, BCH**, and *aRM** algebras) satisfying the identity (x → 1) → y = (y → 1) → x, which plays a very important role in the theory of p-semisimple 96 ANDRZEJ WALENDZIAK BCI algebras. We describe the connections of considered algebras and various generalizations of commutative groups (such as, for example, involutive commutative moons and commutative goops, which were introduced by A. Iorgulescu [5]). In particular, we prove that RM algebras verifying (x → 1) → y = (y → 1) → x are equivalent with involutive commutative moons.

Generalizations of BCI algebras
Let A = (A; →, 1) be an algebra of type (2, 0). We define the binary relation by: for all x, y ∈ A, x y ⇐⇒ x → y = 1.
We consider the following list of properties (cf. [3]) that can be satisfied by A:    7. BCH** algebra if it is a BCH algebra verifying (**).

Generalizations of commutative groups
A. Iorgulescu [5] introduced and studied new generalizations of groups such as moons, goops and many others.
Note that the associative moon is just the group.
Definition 2.6. ( [15]) We say that an algebra (G; ·, −1 , 1) of type (2, 1, 0) is a weakly goop if it is an involutive moon satisfying Let A = (A; →, 1) be an algebra of type (2, 0). Consider the following conditions that can be satisfied by A: Note that, in [5], the concept of negation, −1 , is defined by Thus (D1=) is in fact the double negation property (DN) and (PS) is the property (pDNeg2), in the commutative case, from the book [5].
First we present connections between the conditions in the above list. (iii) Let x, y ∈ A and x y. By (**) and (Re), y → x x → x = 1. From (p-s1) we obtain y → x = 1, that is, y x. Applying (An), we conclude that x = y. Then (p-s) holds.
(iv) This is immediate.
(v) Let x, y ∈ A and suppose that x y. Then x → y = 1. Applying (Ex) and (Re), we get  p-semisimple *aRM** algebra, which is not an RME algebra (hence, it is also not a BCI algebra).   We introduced now the following definition.
(iv) ⇒ (v) By Remark 3.9.    a, a, b). Then A is an aRM(PS) algebra, which is not an *aRM** algebra.
Note that connections between goops and implicative-goops are presented in Chapter 12 of [5].
(iii) Let x, y ∈ G. We have that is, (PS) holds in Ψ(G).