The p-semisimple property for some generalizations of BCI algebras and its applications

. This paper presents some generalizations of BCI algebras (the RM, tRM, *RM, RM**, *RM**, aRM**, *aRM**, BCH**, BZ, pre-BZ and pre-BCI algebras). We investigate the p-semisimple property for algebras mentioned above; give some examples and display various conditions equivalent to p-semisimplicity. Finally, we present a model of mereology without antisymmetry (NAM) which could represent a tRM algebra.


Introduction
The class of all tRM algebras contains BCI, BCK algebras and many others. BCK and BCI algebras had been widely investigated by many authors, but for the first time they were introduced in 1966 by Y. Imai and K. Iséki [5,8], as algebras connected to certain kinds of logics. In 1983, Q. P. Hu and X. Li [4] defined BCH algebras, which contain BCK and BCI algebras; and in 1991, R. Ye [20] introduced the notion of BZ algebra. Recently, A. Iorgulescu [6] has defined fourteen new distinct generalizations of BCI algebras; in particular, *RM, RM**, pre-BZ and pre-BCI algebras, which 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). In [10], T. Lei and C. Xi defined p-semisimple BCI algebras. These algebras have been extensively investigated in many papers [1,3,9,13,18,21], etc. [80]

Lidia Obojska and Andrzej Walendziak
In this manuscript, we would like to investigate the p-semisimple property for the above mentioned algebras. Section 2 will be dedicated to various aspects of BCI, BCK algebras needed to understand the results of our paper. In Section 3, which is the core of this work, we will give several examples of p-semisimple proper RM algebras, *RM algebras, *RM** algebras, pre-BZ algebras, etc. Moreover, various characterizations of p-semisimple algebras will be given. In Section 4 we will review the foundations of non-antisymmetric mereology and represent it as a tRM algebra.

Lemma 1 ([6])
Let A = (A; →, 1) be an algebra of type (2,0). Then the following statements hold: An algebra A = (A; →, 1) is a BCH algebra if (Re), (Ex) and (An) are fulfilled, while a BCH algebra A is a BCI algebra if (B) holds. A BCK algebra is a BCI algebra satisfying (L). Additionally, an algebra A is an RM algebra if (Re) and (M) hold. By Lemma 1 (i), BCH, BCI and BCK algebras are particular cases of RM algebras. We now recall definitions of types of algebras that we will be working with.
• A tRM algebra is an RM algebra satisfying (Tr).
• A pre-BZ algebra is an RM algebra satisfying (B).
• A BZ algebra (also called a weak BCC algebra) is a pre-BZ algebra satisfying (An).
• A pre-BCI algebra is a pre-BZ algebra satisfying (Ex).

Lidia Obojska and Andrzej Walendziak
Now, let us define a binary relation " " as follows It can be verified that in BCK, BCI, BZ and BCH** algebras " " is an order relation; instead in tRM, *RM, RM**, *RM**, pre-BZ and pre-BCI algebras, " " is only reflexive and transitive, i.e. it is a pre-order relation. Moreover, from [7] we recall that • A proper *RM** algebra is a *RM** algebra such that (Ex), (An), (L), (B), (BB) are not satisfied.
• A proper pre-BZ algebra is a pre-BZ algebra such that (BB), (Ex), (An), (L) are not satisfied.
• A proper BZ algebra is a BZ algebra such that (BB), (Ex), (L) are not satisfied.
• A proper pre-BCI algebra is a pre-BCI algebra such that (An), (L) are not satisfied.
• A proper BCI algebra is a BCI algebra such that (L) is not satisfied.
The p-semisimple property for some generalizations of BCI algebras

[83]
Proposition 2 If A is a tRM algebra, then "≈" is an equivalence relation on A.

Proposition 3
If A is a *RM** algebra, then "≈" is a congruence relation on A.
Proof. By Proposition 2, "≈" is an equivalence relation on A. Let x, y, a ∈ A and x ≈ y.
Consequently, "≈' is a congruence relation on A.

Lemma 2
If A is a *RM** algebra and x, y ∈ A, then Proof. We have

Proposition 4
If A is a *RM** algebra, then A ≈ is a *aRM** algebra.
Proof. Obviously, A ≈ verifies (Re) and (M). Let x, y, z ∈ A and y ≈ z ≈ . By Lemma 2, we have y z. Since (*) holds in A, we obtain x → y x → z.
By Proposition 4, the following corollary follows.

Lidia Obojska and Andrzej Walendziak
Example 1 Consider the set A = {a, b, c, 1} and the operation "→" given by the following table.
→ a b c 1 We can observe that properties (Re) It is easy to see that properties (Re), (M) and (*) (hence (Tr)) are satisfied; (An) is not satisfied for (x, y) = (a, b); (B) and (BB) are not satisfied for (x, y, z) The p-semisimple property for some generalizations of BCI algebras   (1) x y =⇒ x ≈ y.
(ii). This proof is obvious from (i).
Observe that (ii) follows from (i).
The p-semisimple property for some generalizations of BCI algebras
(3) =⇒ (p-S). This follows from the proof of Theorem 1. The proof of (i) is complete.
Notice that (ii) follows from (i).
A RM algebra is said to be a BF algebra ([19]) if it satisfies (6 ). By Theorem 2 (ii) we obtain the following corollary.

Corollary 2
Every p-semisimple BZ algebra is a BF algebra.
Observe that (ii) follows from (i).

Theorem 6 Let A be an RM algebra. A is a p-semisimple pre-BZ algebra if and only if
Proof. Let A be a p-semisimple pre-BZ algebra. Applying (B), we have Hence, from (1) we get (B). Conversely, suppose that (B) holds in A. Then A satisfies (B), and hence A is a pre-BZ algebra. Taking z = y in (B), we obtain Therefore, (6) holds in A. By the use of Theorem 2 (i), we conclude that A is p-semisimple.

Corollary 4 Let A be an RM algebra. A is a p-semisimple BZ algebra if and only if
for all x, y, z ∈ A.
Proof. Let A be a p-semisimple aRM** algebra, and x, y, z ∈ A. Suppose that y z, then by Proposition 5 y = z, and x → y = x → z. Thus (*) holds in A. Corollary 5 p-semisimple aRM** algebras coincide with p-semisimple *aRM** algebras.

Proposition 6
Let A be an RM algebra. A is a p-semisimple *aRM** algebra if and only if it satisfies (1 ).
(c) =⇒ (a). It is easy to see that (An), (*) and (**) hold in A. Applying Lemma 1 (vi), we deduce that A satisfies (B), and therefore, A is a BCI algebra. By the use of Proposition 5 (ii), A is p-semisimple.

Some remarks on classical mereology
The invention of mereology was the answer for the set theoretical paradox discovered by B. Russell in foundations of mathematics. Mereology is a set theory based on Leśniewski's concept of a set [11]. It is also called a collective set theory because a set is conceived as an aggregate in which all its parts are interrelated. Mereological objects are assumed to be real; therefore, it is a set theory created for physical objects.
In comparison to classical (Zermelo-Fraenkel) set theory, mereology opens a new perspective of conceiving sets, i.e. a set is conceived as a whole and only from the perspective of the whole do we then distinguish elements. Let I 1 = [0, 1]. Mereologically, it can be considered as a set composed of two halves: S 1 = [0, 1/2], S 2 = [1/2, 1], therefore I 1 = {S 1 , S 2 } or as a set composed of four quarters: /4, 1]. In the case of halves, this set has two elements: S 1 , S 2 ; in the case of quarters, the given set has four elements: Q 1 , Q 2 , Q 3 , Q 4 . Since in mereology the concept of a class is synonymous with the concept of sum, we have Thus, the novelty of mereology provides a new way of conceiving objects: there are wholes which can be divided into parts in different ways, as exemplified by the decay of a physical particle meson π [16].
The extensionality axiom provides a sufficient condition for two sets to be equal. Mereological Extensionality Axiom (MEA), which is slightly different from the ZFC extensionality principle, is expressed as follows It states that two objects having all the same proper subsets are equal. We can observe that in the example of intervals I 1 , I 2 , MEA holds. Moreover, MEA is also conserved for ZFC sets; in fact, I 2 has different subsets than I 1 , therefore I 1 = I 2 . This difference between the ZFC extensionality axiom and the mereological extensionality axiom follows from the fact that a mereological set is not uniquely determined by its elements. In [15] it was shown that the mereological extensionality axiom follows from the antisymmetry of the inclusion relation "⊆". Hence, it is the antisymmetry property for the inclusion relation which is crucial for ZFC set theory and mereology. Now, let us present a classical extensional mereology, formally † . Let M be a universe of objects and "⊆" -the binary relation of being an ingredient. It is assumed that the relation "⊆" partially orders M , ∀ x, y, z ∈ M (x ⊆ y ∧ y ⊆ z =⇒ x ⊆ z).
With the use of the relation "⊆" we may define three auxiliary relations: the relation of overlapping -"•", disjointedness -" " and the relation "⊂", defined on the M × M in the following way Apart from conditions (M1)-((M3), in classical Mereology the so-called Strong Supplementation Principle is assumed ∀ x, y ∈ M (x y =⇒ ∃ z ∈ M (z ⊆ x ∧ z y)). (M4)

Algebraic representations of NAM
Let M be a nonvoid subset of Z and "|" be the relation of divisibility on M . Clearly, "|" satisfies (NAM1) and (NAM3), that is, (M ; |) is a pre-ordered set. Moreover, if M contains elements a and b such that a|b, b|a and a = b (for example, 1 and −1 belong to M ), then M also satisfies (NAM2). The property (L) does not hold for x = 2. Therefore, (M ; →, 1) is a p-semisimple proper *aRM** algebra -an example of an algebra that is not considered in Section 3.

Example 9
Let (G; * , 1) be a group and "→" will be defined on G as follows Then (G; →, 1) is a p-semisimple tRM algebra. Properties (Re), (M), (Tr) are fulfilled; other properties depend on G.