On Symmetric Left Bi-Derivations in BCI-Algebras

Abstract

The notion of symmetric left bi-derivation of a BCI-algebra X is introduced, and related properties are investigated. Some results on componentwise regular and d-regular symmetric left bi-derivations are obtained. Finally, characterizations of a p-semisimple BCI-algebra are explored, and it is proved that, in a p-semisimple BCI-algebra, F is a symmetric left bi-derivation if and only if it is a symmetric bi-derivation.

1. Introduction

BCK-algebras and BCI-algebras are two classes of nonclassical logic algebras which were introduced by Imai and Iséki in 1966 [1, 2]. They are algebraic formulation of BCK-system and BCI-system in combinatory logic. Later on, the notion of BCI-algebras has been extensively investigated by many researchers (see [3–6], and references therein). The notion of a BCI-algebra generalizes the notion of a BCK-algebra in the sense that every BCK-algebra is a BCI-algebra but not vice versa (see [7]). Hence, most of the algebras related to the𝑡-norm-based logic such as MTL [8], BL, hoop, MV [9] (i.e lattice implication algebra), and Boolean algebras are extensions of BCK-algebras (i.e. they are subclasses of BCK-algebras) which have a lot of applications in computer science (see [10]). This shows that BCK-/BCI-algebras are considerably general structures.

Throughout our discussion,𝑋will denote a BCI-algebra unless otherwise mentioned. In the year 2004, Jun and Xin [11] applied the notion of derivation in ring and near-ring theory to BCI-algebras, and as a result they introduced a new concept, called a (regular) derivation, in BCI-algebras. Using this concept as defined they investigated some of its properties. Using the notion of a regular derivation, they also established characterizations of a𝑝-semisimple BCI-algebra. For a self-map𝑑of a BCI-algebra, they defined a𝑑-invariant ideal and gave conditions for an ideal to be𝑑-invariant. According to Jun and Xin, a self map𝑑∶𝑋→𝑋is called a left-right derivation (briefly(𝑙,𝑟)-derivation) of𝑋if𝑑(𝑥∗𝑦)=𝑑(𝑥)∗𝑦∧𝑥∗𝑑(𝑦)holds for all𝑥,𝑦∈𝑋. Similarly, a self map𝑑∶𝑋→𝑋is called a right-left derivation (briefly(𝑟,𝑙)-derivation) of𝑋if𝑑(𝑥∗𝑦)=𝑥∗𝑑(𝑦)∧𝑑(𝑥)∗𝑦holds for all𝑥,𝑦∈𝑋. Moreover, if𝑑is both(𝑙,𝑟)- and(𝑟,𝑙)-derivation, it is a derivation on𝑋. After the work of Jun and Xin [11], many research articles have appeared on the derivations of BCI-algebras and a greater interest has been devoted to the study of derivations in BCI-algebras on various aspects (see [12–17]).

Inspired by the notions of𝜎-derivation [18], left derivation [19], and symmetric bi-derivations [20, 21] in rings and near-rings theory, many authors have applied these notions in a similar way to the theory of BCI-algebras (see [12, 13, 17]). For instantce in 2005 [17], Zhan and Liu have given the notion of𝑓-derivation of BCI-algebras as follows: a self map𝑑_𝑓∶𝑋→𝑋is said to be a left-right𝑓-derivation or(𝑙,𝑟)-𝑓-derivation of𝑋if it satisfies the identity𝑑_𝑓(𝑥∗𝑦)=𝑑_𝑓(𝑥)∗𝑓(𝑦)∧𝑓(𝑥)∗𝑑_𝑓(𝑦)for all𝑥,𝑦∈𝑋. Similarly, a self map𝑑_𝑓∶𝑋→𝑋is said to be a right-left𝑓-derivation or(𝑟,𝑙)-𝑓-derivation of𝑋if it satisfies the identity𝑑_𝑓(𝑥∗𝑦)=𝑓(𝑥)∗𝑑_𝑓(𝑦)∧𝑑_𝑓(𝑥)∗𝑓(𝑦)for all𝑥,𝑦∈𝑋. Moreover, if𝑑_𝑓is both(𝑙,𝑟)- and(𝑟,𝑙)-𝑓-derivation, it is said that𝑑_𝑓is an𝑓-derivation, where𝑓is an endomorphism. In the year 2007, Abujabal and Al-Shehri [12] defined and studied the notion of left derivation of BCI-algebras as follows: a self map𝐷∶𝑋→𝑋is said to be a left derivation of𝑋if satisfying𝐷(𝑥∗𝑦)=𝑥∗𝐷(𝑦)∧𝑦∗𝐷(𝑥)for all𝑥,𝑦∈𝑋. Furthermore, in 2011 [13], Ilbira et al. have introduced the notion of symmetric bi-derivations in BCI-algebras. Following [13], a mapping𝐷(⋅,⋅)∶𝑋×𝑋→𝑋is said to be symmetric if𝐹(𝑥,𝑦)=𝐹(𝑦,𝑥)holds for all pairs𝑥,𝑦∈𝑋.A symmetric mapping𝐷(⋅,⋅)∶𝑋×𝑋→𝑋is called left-right symmetric bi-derivation (briefly(𝑙,𝑟)-symmetric bi-derivation) if it satisfies the identity𝐷(𝑥∗𝑦,𝑧)=𝐷(𝑥,𝑧)∗𝑦∧𝑥∗𝐷(𝑦,𝑧)for all𝑥,𝑦,𝑧∈𝑋.𝐷is called right-left symmetric bi-derivation (briefly(𝑟,𝑙)-symmetric bi-derivation) if it satisfies the identity𝐷(𝑥∗𝑦,𝑧)=𝑥∗𝐷(𝑦,𝑧)∧𝐷(𝑥,𝑧)∗𝑦for all𝑥,𝑦,𝑧∈𝑋. Moreover, if𝐷is both a(𝑙,𝑟)- and a(𝑟,𝑙)-symmetric bi-derivation, it is said that𝐷is a symmetric bi-derivation on𝑋.

Motivated by the notion of symmetric bi-derivations [13] in the theory of BCI-algebras, in the present analysis, we introduced the notion of symmetric left bi-derivations on BCI-algebras and investigated related properties. Further, we obtain some results on componentwise regular and𝑑-regular symmetric left bi-derivations. Finally, we characterize the notion of𝑝-semisimple BCI-algebra𝑋by using the concept of symmetric left bi-derivation and show that, in a 𝑝-semisimple BCI-algebra𝑋,𝐹is a symmetric left bi-derivation if and only if it is a symmetric bi-derivation.

2. Preliminaries

We begin with the following definitions and properties that will be needed in the sequel.

A nonempty set𝑋with a constant0and a binary operation∗is called a BCI-algebra if for all𝑥,𝑦,𝑧∈𝑋the following conditions hold: (I)((𝑥∗𝑦)∗(𝑥∗𝑧))∗(𝑧∗𝑦)=0,(II)(𝑥∗(𝑥∗𝑦))∗𝑦=0,(III)𝑥∗𝑥=0,(IV)𝑥∗𝑦=0and𝑦∗𝑥=0imply𝑥=𝑦.

Define a binary relation≤on𝑋by letting𝑥∗𝑦=0if and only if𝑥≤𝑦. Then(𝑋,≤)is a partially ordered set. A BCI-algebra𝑋satisfying0≤𝑥for all𝑥∈𝑋 is called BCK-algebra.

A𝐵𝐶𝐼-algebra𝑋has the following properties for all𝑥,𝑦,𝑧∈𝑋.(a1)𝑥∗0=𝑥. (a2)(𝑥∗𝑦)∗𝑧=(𝑥∗𝑧)∗𝑦.(a3)𝑥≤𝑦implies𝑥∗𝑧≤𝑦∗𝑧and𝑧∗𝑦≤𝑧∗𝑥. (a4)(𝑥∗𝑧)∗(𝑦∗𝑧)≤𝑥∗𝑦.(a5)𝑥∗(𝑥∗(𝑥∗𝑦))=𝑥∗𝑦.(a6)0∗(𝑥∗𝑦)=(0∗𝑥)∗(0∗𝑦).(a7)𝑥∗0=0implies𝑥=0.

For a BCI-algebra𝑋, denote by𝑋₊(resp., 𝐺(𝑋)) the𝐵𝐶𝐾-part (resp., the𝐵𝐶𝐼-G part) of𝑋; that is,𝑋₊is the set of all𝑥∈𝑋such that0≤𝑥(resp., 𝐺(𝑋)∶={𝑥∈𝑋∣0∗𝑥=𝑥}). Note that𝐺(𝑋)∩𝑋₊={0}(see [22]). If𝑋₊={0}, then𝑋is called a𝑝-semisimple 𝐵𝐶𝐼-algebra. In a𝑝-semisimple𝐵𝐶𝐼-algebra𝑋, the following hold. (a8)(𝑥∗𝑧)∗(𝑦∗𝑧)=𝑥∗𝑦. (a9)0∗(0∗𝑥)=𝑥for all𝑥∈𝑋. (a10)𝑥∗(0∗𝑦)=𝑦∗(0∗𝑥). (a11)𝑥∗𝑦=0implies𝑥=𝑦. (a12)𝑥∗𝑎=𝑥∗𝑏implies𝑎=𝑏. (a13)𝑎∗𝑥=𝑏∗𝑥implies𝑎=𝑏. (a14)𝑎∗(𝑎∗𝑥)=𝑥. (a15)(𝑥∗𝑦)∗(𝑤∗𝑧)=(𝑥∗𝑤)∗(𝑦∗𝑧).

Let𝑋be a𝑝-semisimple BCI-algebra. We define addition “+” as𝑥+𝑦=𝑥∗(0∗𝑦)for all𝑥,𝑦∈𝑋. Then(𝑋,+)is an abelian group with identity0and𝑥−𝑦=𝑥∗𝑦. Conversely, let(𝑋,+)be an abelian group with identity0, and let𝑥∗𝑦=𝑥−𝑦. Then𝑋is a𝑝-semisimple BCI-algebra and𝑥+𝑦=𝑥∗(0∗𝑦)for all𝑥,𝑦∈𝑋(see [6]).

For a BCI-algebra𝑋, we denote𝑥∧𝑦=𝑦∗(𝑦∗𝑥), in particular0∗(0∗𝑥)=𝑎_𝑥, and𝐿_𝑝(𝑋)∶={𝑎∈𝑋∣𝑥∗𝑎=0⇒𝑥=𝑎,∀𝑥∈𝑋}.We call the elements of𝐿_𝑝(𝑋)the𝑝-atoms of𝑋. For any𝑎∈𝑋, let𝑉(𝑎)∶={𝑥∈𝑋∣𝑎∗𝑥=0}, which is called the branch of𝑋with respect to𝑎. It follows that𝑥∗𝑦∈𝑉(𝑎∗𝑏)whenever𝑥∈𝑉(𝑎)and𝑦∈𝑉(𝑏)for all𝑥,𝑦∈𝑋and all𝑎,𝑏∈𝐿_𝑝(𝑋). Note that𝐿_𝑝(𝑋)={𝑥∈𝑋∣𝑎_𝑥=𝑥},which is the𝑝-semisimple part of𝑋, and𝑋is a𝑝-semisimple BCI-algebra if and only if𝐿_𝑝(𝑋)=𝑋(see [23, Proposition 3.2]). Note also that𝑎_𝑥∈𝐿_𝑝(𝑋); that is,0∗(0∗𝑎_𝑥)=𝑎_𝑥, which implies that𝑎_𝑥∗𝑦∈𝐿_𝑝(𝑋)for all𝑦∈𝑋. It is clear that𝐺(𝑋)⊂𝐿_𝑝(𝑋), and𝑥∗(𝑥∗𝑎)=𝑎and𝑎∗𝑥∈𝐿_𝑝(𝑋)for all𝑎∈𝐿_𝑝(𝑋)and all𝑥∈𝑋. Let𝐷(⋅,⋅)∶𝑋×𝑋→𝑋be a symmetric mapping. Then for all𝑥∈𝑋, a mapping𝑑∶𝑋→𝑋defined by𝑑(𝑥)=𝐷(𝑥,𝑥)is called trace of𝐷[13]. For more details, refer to [3, 4, 6, 11, 22, 23].

3. Symmetric Left Bi-Derivations

The following definition introduces the notion of symmetric left bi-derivation for a BCI-algebra𝑋.

Definition 1. A symmetric mapping𝐹(⋅,⋅)∶𝑋×𝑋→𝑋is called a symmetric left bi-derivation of𝑋if it satisfies the following identity: 𝐹=∧().∀𝑥,𝑦,𝑧∈𝑋𝑥∗𝑦,𝑧𝑥∗𝐹𝑦,𝑧𝑦∗𝐹𝑥,𝑧 (1)

Example 2 (see [24]). Consider a𝑝-semisimple BCI-algebra𝑋={0,3,4,5}with the following Cayley table: ∗034503034543054545035430 (2)
Define a mapping𝐹(⋅,⋅)∶𝑋×𝑋→𝑋by 𝐹()()()()𝐹()()𝐹()()𝐹()()𝐹()()𝐹()()𝐹()()0,0=𝐹3,3=𝐹4,4=𝐹5,5=0,0,3=𝐹3,0=3,0,4=𝐹4,0=4,0,5=𝐹5,0=5,3,4=𝐹4,3=5,3,5=𝐹5,3=4,4,5=𝐹5,4=3. (3)
It is routine to verify that𝐹is a symmetric left bi-derivation of𝑋.

Theorem 3. Let𝐹(⋅,⋅)∶𝑋×𝑋→𝑋be a symmetric left bi-derivation of𝑋.Then (1)(∀𝑧∈𝑋)(𝑎∈𝐺(𝑋)⇒𝐹(𝑎,𝑧)∈𝐺(𝑋)).(2)(∀𝑧∈𝑋)(𝑎∈𝐿_𝑝(𝑋)⇒𝐹(𝑎,𝑧)∈𝐿_𝑝(𝑋)).(3)(∀𝑧∈𝑋)(𝑎∈𝐿_𝑝(𝑋)⇒𝐹(𝑎,𝑧)=0+𝐹(𝑎,𝑧)).(4)(∀𝑧∈𝑋)(𝑎∈𝐿_𝑝(𝑋)⇒𝐹(𝑎,𝑧)=𝑎∗𝐹(0,𝑧)=𝑎+𝐹(0,𝑧)).

Proof. (1) Let𝑎∈𝐺(𝑋).Then0∗𝑎=𝑎, and so 𝐹()()=((∧((=((∗(∗(((),𝑎,𝑧=𝐹0∗𝑎,𝑧0∗𝐹𝑎,𝑧))𝑎∗𝐹0,𝑧))𝑎∗𝐹0,𝑧))((𝑎∗𝐹0,𝑧))0∗𝐹𝑎,𝑧)))=0∗𝐹𝑎,𝑧 (4) since0∗𝐹(𝑎,𝑧)∈𝐿_𝑝(𝑋).Hence𝐹(𝑎,𝑧)∈𝐺(𝑋).
(2) For any𝑎∈𝐿_𝑝(𝑋)implies𝑎=0∗(0∗𝑎)and so 𝐹()(())=((∧)(=)(∗)(∗((()𝑎,𝑧=𝐹0∗0∗𝑎,𝑧0∗𝐹0∗𝑎,𝑧))((0∗𝑎∗𝐹0,𝑧))((0∗𝑎∗𝐹0,𝑧))(((0∗𝑎∗𝐹0,𝑧))0∗𝐹0∗𝑎,𝑧)))=0∗𝐹0∗𝑎,𝑧∈𝐿_𝑝(𝑋). (5)
(3) By (2), we have𝐹(𝑎,𝑧)∈𝐿_𝑝(𝑋). Then 𝐹(𝑎,𝑧)=0∗(0∗𝐹(𝑎,𝑧))=0+𝐹(𝑎,𝑧). (6)
(4) For any𝑎∈𝐿_𝑝(𝑋)and𝑧∈𝑋, we have 𝐹()()=((∧((=((∗(∗((()((().𝑎,𝑧=𝐹𝑎∗0,𝑧𝑎∗𝐹0,𝑧))0∗𝐹𝑎,𝑧))0∗𝐹𝑎,𝑧))((0∗𝐹𝑎,𝑧))𝑎∗𝐹0,𝑧)))=𝑎∗𝐹0,𝑧=𝑎∗0∗𝐹0,𝑧))=𝑎+𝐹0,𝑧 (7) This completes the proof.

Using Theorem 3, we have the following corollary.

Corollary 4. Let𝐹(⋅,⋅)∶𝑋×𝑋→𝑋be a symmetric left bi-derivation and𝑑∶𝑋→𝑋be the trace of𝐹.Then (1)(∀𝑎∈𝐺(𝑋))(𝑑(𝑎)∈𝐺(𝑋)).(2)(∀𝑎∈𝐿_𝑝(𝑋))(𝑑(𝑎)∈𝐿_𝑝(𝑋)).

Theorem 5. Let𝐹be a symmetric left bi-derivation of𝑋.Then (1)(∀𝑧∈𝑋)(𝑎,𝑏∈𝐿_𝑝(𝑋)⇒𝐹(𝑎+𝑏,𝑧)=𝑎+𝐹(𝑏,𝑧)).(2)(∀𝑧∈𝑋)(𝑎∈𝐿_𝑝(𝑋)⇒𝐹(𝑎,𝑧)=𝑎ifandonlyif𝐹(0,𝑧)=0).(3)(∀𝑥,𝑦,𝑧∈𝑋)(𝐹(𝑥∗𝑦,𝑧)≤𝑥∗𝐹(𝑦,𝑧)).(4)(∀𝑥,𝑦,𝑧∈𝑋)(𝑥∗𝐹(𝑥,𝑧)=𝑦∗𝐹(𝑦,𝑧)).

Proof. (1) Let𝑎,𝑏∈𝐿_𝑝(𝑋). Then 𝐹()(())()∧()()()(()(((().𝑎+𝑏,𝑧=𝐹𝑎∗0∗𝑏,𝑧=𝑎∗𝐹0∗𝑏,𝑧0∗𝑏∗𝐹𝑎,𝑧=𝑎∗𝐹0∗𝑏,𝑧=𝑎∗0∗𝐹𝑏,𝑧∧𝑏∗𝐹0,𝑧))=𝑎∗0∗𝐹𝑏,𝑧))=𝑎+𝐹𝑏,𝑧 (8)
(2) Suppose𝐹(𝑎,𝑧)=𝑎for all𝑎∈𝐿_𝑝(𝑋),𝑧∈𝑋. It is clear that, for0∈𝐿_𝑝(𝑋), we have𝐹(0,𝑧)=0.Conversely let us assume that𝐹(0,𝑧)=0;then by using Theorem 3(4), we have𝐹(𝑎,𝑧)=𝑎+𝐹(0,𝑧)=𝑎+0=𝑎.
(3) For any𝑥,𝑦,𝑧∈𝑋, we have 𝐹=∧()=()∗()∗.𝑥∗𝑦,𝑧𝑥∗𝐹𝑦,𝑧𝑦∗𝐹𝑥,𝑧𝑦∗𝐹𝑥,𝑧𝑦∗𝐹𝑥,𝑧𝑥∗𝐹𝑦,𝑧≤𝑥∗𝐹𝑦,𝑧 (9)
(4) For any 𝑥,𝑧∈𝑋, we have 𝐹()()=((∧((().0,𝑧=𝐹𝑥∗𝑥,𝑧𝑥∗𝐹𝑥,𝑧))𝑥∗𝐹𝑥,𝑧))=𝑥∗𝐹𝑥,𝑧 (10) Thus, we can write𝐹(0,𝑧)=𝑥∗𝐹(𝑥,𝑧)=𝑦∗𝐹(𝑦,𝑧)for any𝑦∈𝑋.This completes the proof.

Definition 6. A symmetric left bi-derivation 𝐹(⋅,⋅)∶𝑋×𝑋→𝑋of a BCI-algebra 𝑋 is said to be componentwise regular if 𝐹(0,𝑧)=0 for all 𝑧∈𝑋. In particular, 𝐹 is called 𝑑-regular if 𝐹(0,0)=𝑑(0)=0.

Theorem 7. Let𝐹be a symmetric left bi-derivation of BCI-algebra𝑋. Then𝑋is a BCK-algebra if and only if𝐹is componentwise regular symmetric left bi-derivation.

Proof. Suppose𝑋is a BCK-algebra. Then for any𝑥,𝑧∈𝑋, we have 𝐹()()=((∧((((0,𝑧=𝐹0∗𝑥,𝑧0∗𝐹𝑥,𝑧))𝑥∗𝐹0,𝑧))=0∧𝑥∗𝐹0,𝑧))=0. (11) Hence𝐹is componentwise regular.
Conversely, let𝐹be a componentwise regular symmetric left bi-derivation. Let for any𝑎∈𝐿_𝑝(𝑋)be such that𝑎≠0. Then 𝐹(𝑎∗0,0)=𝐹(𝑎,0)=0. (12) But it is clear that ()()()𝑎∗𝐹0,0∧0∗𝐹𝑎,0=𝑎∗0∧0∗0=𝑎∧0=0∗0∗𝑎=𝑎≠0, (13) which is not possible as𝐹is a componentwise regular symmetric left bi-derivation. Thus0is the unique𝑝-atom. Assume that for some𝑚∈𝑋, we have0∗𝑚≠0, then𝑎_0∗𝑚=0∗(0∗(0∗𝑚))=0, so 0∗𝑚∈𝐿_𝑝(𝑋), which is a contradiction. Henceforth, for all𝑚∈𝑋, we have0∗𝑚=0implying thereby,𝑋is a BCK-algebra.
This completes the proof.

Theorem 8. Let𝐹be a componentwise regular symmetric left bi-derivation of a BCI-algebra𝑋.Then (1)Both𝑥and𝐹(𝑥,𝑧)belong to the same branch for all𝑥,𝑧∈𝑋.(2)(∀𝑥,𝑧∈𝑋)(𝐹(𝑥,𝑧)≤𝑥).(3)(∀𝑥,𝑦,𝑧∈𝑋)(𝐹(𝑥,𝑧)∗𝑦≤𝑥∗𝐹(𝑦,𝑧)).

Proof. (1) For any𝑥,𝑧∈𝑋, we get 0=𝐹(0,𝑧)=𝐹𝑎_𝑥=𝑎∗𝑥,𝑧_𝑥∗𝐹(𝑥,𝑧)∧𝑥∗𝐹𝑎_𝑥=𝑎,𝑧𝑥∗𝐹_𝑥,𝑧∗𝑥∗𝐹𝑎_𝑥,𝑧∗𝑎_𝑥∗𝐹(𝑥,𝑧)=𝑎_𝑥∗𝐹(𝑥,𝑧), (14) since𝑎_𝑥∗𝐹(𝑥,𝑧)∈𝐿_𝑝(𝑋).Hence𝑎_𝑥≤𝐹(𝑥,𝑧),and so𝐹(𝑥,𝑧)∈𝑉(𝑎_𝑥).Obviously,𝑥∈𝑉(𝑎_𝑥).
(2) Since𝐹is componentwise regular,𝐹(0,𝑧)=0.Then 𝐹()()=((∧((=()∧((=((∗()𝑥,𝑧=𝐹𝑥∗0,𝑧𝑥∗𝐹0,𝑧))0∗𝐹𝑥,𝑧))𝑥∗00∗𝐹𝑥,𝑧))0∗𝐹𝑥,𝑧))((0∗𝐹𝑥,𝑧))∗𝑥≤𝑥. (15)
(3) Since𝐹(𝑥,𝑧)≤𝑥for all𝑥,𝑧∈𝑋by (2), using (a3) we obtain 𝐹(𝑥,𝑧)∗𝑦≤𝑥∗𝑦≤𝑥∗𝐹𝑦,𝑧. (16) This completes the proof.

Next, we prove some results in a𝑝-semisimple BCI-algebra.

Theorem 9. Let𝐹be a symmetric left bi-derivation of a𝑝-semisimple BCI-algebra𝑋; one has the following assertions.(1)(∀𝑥,𝑦,𝑧∈𝑋)(𝐹(𝑥∗𝑦,𝑧)=𝑥∗𝐹(𝑦,𝑧)).(2)(∀𝑥,𝑦,𝑧∈𝑋)(𝐹(𝑥,𝑧)∗𝑥=𝐹(𝑦,𝑧)∗𝑦).(3)(∀𝑥,𝑦,𝑧∈𝑋)(𝐹(𝑥,𝑧)∗𝑥=𝑦∗𝐹(𝑦,𝑧)).

Proof. (1) Let𝑋be a𝑝-semisimple BCI-algebra. Then for any𝑥,𝑦,𝑧∈𝑋, we have 𝐹𝑥∗𝑦,𝑧=𝑥∗𝐹𝑦,𝑧∧𝑦∗𝐹(𝑥,𝑧)=𝑥∗𝐹𝑦,𝑧. (17)
(2) Let𝑥,𝑦,𝑧∈𝑋. Using (I), we have ∗∗()()𝑥∗𝑦𝑥∗𝐹𝑦,𝑧≤𝐹𝑦,𝑧∗𝑦,𝑦∗𝑥𝑦∗𝐹𝑥,𝑧≤𝐹𝑥,𝑧∗𝑥. (18) These above inequalities can be rewritten as ∗∗𝐹∗()∗(𝐹())𝑥∗𝑦𝑥∗𝐹𝑦,𝑧𝑦,𝑧∗𝑦=0,𝑦∗𝑥𝑦∗𝐹𝑥,𝑧𝑥,𝑧∗𝑥=0. (19) Consequently, we get ∗∗𝐹=∗()∗(𝐹())𝑥∗𝑦𝑥∗𝐹𝑦,𝑧𝑦,𝑧∗𝑦𝑦∗𝑥𝑦∗𝐹𝑥,𝑧𝑥,𝑧∗𝑥 (20) Also, using Theorem 5(4) and (1), we obtain =⟹∗=∗().𝑥∗𝑦∗𝐹𝑥∗𝑦,𝑧𝑦∗𝑥∗𝐹𝑦∗𝑥,𝑧𝑥∗𝑦𝑥∗𝐹𝑦,𝑧𝑦∗𝑥𝑦∗𝐹𝑥,𝑧 (21)
Since𝑋is a𝑝-semisimple BCI-algebra, hence, by using (21) and (a12), the above (20) yields𝐹(𝑥,𝑧)∗𝑥=𝐹(𝑦,𝑧)∗𝑦.
(3) We have𝐹(0,𝑧)=𝑥∗𝐹(𝑥,𝑧)by Theorem 5(4). Further, on letting𝑥=0, we get that 𝐹(0,𝑧)∗0=𝐹(𝑦,𝑧)∗𝑦implies𝐹(0,𝑧)=𝐹(𝑦,𝑧)∗𝑦. Henceforth𝐹(𝑦,𝑧)∗𝑦=𝑥∗𝐹(𝑥,𝑧), which amounts to say that𝐹(𝑥,𝑧)∗𝑥=𝑦∗𝐹(𝑦,𝑧).
This completes the proof.

Theorem 10. Let𝑋be a𝑝-semisimple BCI-algebra. Then𝐹is a symmetric left bi-derivation if and only if it is a symmetric bi-derivation on𝑋.

Proof. Suppose that𝐹is a symmetric left bi-derivation on𝑋.First, we show that𝐹is a (r,l)-symmetric bi-derivation on𝑋. Let𝑥,𝑦,𝑧∈𝑋. Using Theorem 9(1) and (a14), we have 𝐹=𝐹()∗𝐹()∗=∧𝐹().𝑥∗𝑦,𝑧=𝑥∗𝐹𝑦,𝑧𝑥,𝑧∗𝑦𝑥,𝑧∗𝑦𝑥∗𝐹𝑦,𝑧𝑥∗𝐹𝑦,𝑧𝑥,𝑧∗𝑦 (22) Hence𝐹is a(𝑟,𝑙)-symmetric bi-derivation on𝑋.
Again, we show that𝐹is a (l,r)-symmetric bi-derivation on𝑋. Let𝑥,𝑦,𝑧∈𝑋. Using Theorem 9(1), (3) and (a15), we have 𝐹=()=((𝐹()(=((∗𝐹=∗((∗𝐹=∗∗𝐹()=𝐹()∧.𝑥∗𝑦,𝑧=𝑥∗𝐹𝑦,𝑧𝑥∗0∗𝐹𝑦,𝑧𝑥∗0,𝑧∗𝐹0,𝑧)))∗𝐹𝑦,𝑧𝑥∗𝑥∗𝐹𝑥,𝑧))𝑦,𝑧∗𝑦∗𝐹𝑦,𝑧𝑥∗𝐹𝑦,𝑧𝑥∗𝐹𝑥,𝑧))𝑦,𝑧∗𝑦𝑥∗𝐹𝑦,𝑧𝑥∗𝐹𝑦,𝑧𝑥,𝑧∗𝑦𝑥,𝑧∗𝑦𝑥∗𝐹𝑦,𝑧 (23)
Conversely, suppose that𝐹is a symmetric bi-derivation of𝑋.As𝐹is a (r,l)-symmetric bi-derivation on𝑋, then for any𝑥,𝑦,𝑧∈𝑋and using (a14), we have 𝐹=∧𝐹()=𝐹()∗𝐹()∗=()∗()∗=∧().𝑥∗𝑦,𝑧𝑥∗𝐹𝑦,𝑧𝑥,𝑧∗𝑦𝑥,𝑧∗𝑦𝑥,𝑧∗𝑦𝑥∗𝐹𝑦,𝑧=𝑥∗𝐹𝑦,𝑧𝑦∗𝐹𝑥,𝑧𝑦∗𝐹𝑥,𝑧𝑥∗𝐹𝑦,𝑧𝑥∗𝐹𝑦,𝑧𝑦∗𝐹𝑥,𝑧 (24) Hence𝐹is a symmetric left bi-derivation. This completes the proof.

Acknowledgments

The authors are grateful to the anonymous referee(s) for a careful checking of the details and for helpful comments that improved the present paper. G. Muhiuddin and Abdullah M. Al-roqi were partially supported by the Deanship of Scientific Research, University of Tabuk, Ministry of Higher Education, Saudi Arabia.