Some generalizations of p-semisimple BCI algebras and groups

We introduce and investigate the strong p-semisimple property for some generalizations of BCI algebras. For BCI algebras, the strong p-semisimple property is equivalent to the p-semisimple property. We describe the connections of strongly p-semisimple algebras and various generalizations of groups (such as, for example, involutive moons and goops). Moreover, we present some examples of proper strongly p-semisimple algebras.


Introduction
introduced BCI algebras as algebraic models of BCI-logic. Hu and Li (1983) defined BCH algebras, which are a generalization of BCI algebras. Later on, Ye (1991) introduced the notion of BZ algebras. It is known that BCI algebras are contained in the class of BZ algebras. Next, the new class of algebras called BH algebras was introduced by Jun et al. (1998). These algebras are a common generalization of BCH and BZ algebras (hence also a generalization of BCI algebras). Recently, Iorgulescu (2016) introduced new generalizations of BCI algebras such as aRM**, *aRM**, BCH** algebras, and many others. 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). The implicative and commutative properties for some subclasses of the class of RM algebras were studied by Walendziak (2018Walendziak ( , 2019. Lei and Xi (1985) defined p-semisimple BCI algebras and proved that p-semisimple BCI algebras are equivalent with abelian groups. The p-semisimple BCI algebras have been extensively investigated in many papers (for example Meng (1987), Hoo (1990), Aslam and Thaheem (1991) (1991), Kim and Park (2005)). Zhang and Ye (1995) showed that p-semisimple BZ algebras are equivalent with groups.
In this paper, we introduce the notion of strongly psemisimple RM algebra. For BZ and BCI algebras, the strong p-semisimple property is equivalent to the p-semisimple property. We describe the connections of strongly p-semisimple algebras and various generalizations of groups (such as, for example, involutive moons and goops, which were introduced by Iorgulescu (2018)). In particular, we prove that strongly p-semisimple RM algebras are equivalent with involutive moons. Moreover, we give some examples of proper strongly p-semisimple algebras.

Generalizations of BCI algebras
Let A = (A; →, 1) be an algebra of type (2, 0). We consider the following list of properties (cf. Iorgulescu 2016) that can be satisfied by A: Following Iorgulescu (2016), we say that a RM algebra A is an aRM algebra if it satisfies (An). We note that aRM algebras are also called BH algebras [see, for example, Yu et al. (1999), Zhang et al. (2001), Jun et al. (2004)]. Now, we will define the following algebras (Iorgulescu 2016): • An aRM** algebra is an aRM algebra satisfying (**).
By Lemma 2.2(i), BCH** ⊂ aRM**(D1). The interrelationships between the classes of algebras mentioned before are visualized in Figure 1. (An arrow indicates proper inclusion, that is, if X and Y are classes of algebras, then X −→ Y means X ⊂ Y.) Iorgulescu (2018) introduced and studied new generalizations of groups such as moons, goops, and many others.

Definition 2.3 A moon is an algebra
Note that the associative moon is just the group.

) is a goop if and only if it is an involutive moon satisfying
Thus G is an involutive moon. By (GP2), it satisfies (GP).

Definition 2.6
We say that an algebra (G; ·, −1 , 1) of type Let involutive moon, weakly goop, goop, group, and abelian group denote the class of all involutive moons, weakly goops, goops, groups, and abelian groups, respectively. From the definitions we obtain involutive moon ⊂ weakly goop ⊂ goop ⊂ group ⊂ abelian group.

p-semisimple and strongly p-semisimple algebras
Let A = (A; →, 1) be an algebra of type (2, 0). Consider the following properties that can be satisfied by A: Note that, in Iorgulescu (2018), the concept of negation, −1 , is defined by x −1 = x → 1, and hence Thus (D1=) is in fact the double negation property (DN) and (PS) is the property (pDNeg2), in the commutative case, from the book Iorgulescu (2018). Remark that RM algebras satisfying (PS) were studied in Walendziak (2020).
First we present connections between the conditions in the above list.

Lemma 3.1 If an algebra
Proof The proof is immediate.
Note that from Lemma 3.1 it follows that every strongly p-semisimple RM algebra is p-semisimple.  1 a a a  b a 1 a a  c a a 1 a  1 a b c 1 Properties (Re), (M), (An), (*), (**) (hence (Tr)), and (p-s) are satisfied. The algebra A = (A; →, 1) does not satisfy (D1=). Therefore, A is a p-semisimple *aRM** algebra that is not strongly p-semisimple.
Denote by strongly p-s-aRM** the class of all strongly psemisimple aRM** algebras (= the class of all p-semisimple aRM**(D1) algebras = the class of all strongly p-semisimple *aRM** algebras = class of all p-semisimple *aRM**(D1) algebras). Proposition 3.9 Let A = (A; →, 1) be an algebra of type (2, 0). The following are equivalent: Proof Let A be a p-semisimple BCH** algebra. By Proposition 3.2, A satisfies (p-s). From (p-s) we deduce that A also satisfies (*). Applying Lemma 2.1(vii) and (v) we see that (B) holds in A. Consequently, A is a BCI algebra. The converse is obvious.
Denote by p-s-BCI the class of all p-semisimple BCI algebras (= the class of all p-semisimple BCH** algebras). Let p-s-BZ (resp. strongly p-s-RM, strongly p-s-aRM) denote the class of all p-semisimple BZ algebras (resp. strongly p-semisimple RM algebras, strongly p-semisimple aRM algebras).

Connections between RM algebras, moons and goops
In this subsection, we establish the connections between: strongly p-semisimple RM algebras and involutive moons, strongly p-semisimple aRM algebras and weakly goops, strongly p-semisimple aRM** algebras and goops.

Proof
(i) Let A be a strongly p-semisimple aRM algebra. By Theorem 3.10, (A) is an involutive moon. Let x, y ∈ A and suppose that y · x −1 = 1 = x · y −1 . Hence x ≤ y and y ≤ x. Therefore x = y by (An). Then (wGP) holds in (A), that is, (A) is a weakly goop. Let now A be a strongly p-semisimple aRM** algebra. Then A satisfies (p-s1), and also (p-s) by Proposition 3.2. Let y · x −1 = 1, and hence x ≤ y. From (p-s) it follows that x = y . Consequently, (GP) holds in (A), that is, (A) is a goop by Proposition 2.5. (ii) Let G be a weakly goop. By Theorem 3.10, (G) is a strongly p-semisimple RM algebra. From (wGP) we deduce that (An) holds in (G). Thus (G) is a strongly p-semisimple aRM algebra. Let now G be a goop. From (GP) it follows that (p-s) holds in (G). Hence, obviously, (G) satisfies (**). Thus (G) is a strongly p-semisimple aRM** algebra. (iii) See the proof of Theorem 3.10(iii).
Hence, by above Theorem 3.11, we have the equivalences: strongly p-s-aRM ≡ weakly goop, strongly p-s-aRM** ≡ goop, that is, the strongly p-semisimple aRM algebras are term equivalent to the weakly goops and the strongly p-semisimple aRM** algebras are term equivalent to the goops.  From Theorems 3.12 and 3.13 it follows that the RM algebras with (PS) are term equivalent to the commutative involutive moons, the aRM algebras with (PS) are term equivalent to the commutative weakly goops, and the *aRM** algebras with (PS) are term equivalent to the commutative goops.
From Theorems 3.14 and 3.15 we see that p-s-BZ ≡ group and p-s-BCI ≡ abelian group, that is, the p-semisimple BZ algebras are term equivalent to the groups and the p-semisimple BCI algebras are term equivalent to the abelian groups.  1 1 b a  b c 1 c b  c 1 b 1 c  1 a b c 1 .

Examples of proper strongly p-semisimple algebras
The algebra A = (A; →, 1) satisfies properties (An), (Re), (M), and (D1=) (hence (p-s1)). It does not satisfy (Ex) and (Tr) for (x, y, z) = (c, a, b). Then A is a proper strongly p-semisimple aRM algebra. long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/.