Research article

Branchwise solid generalized BCH-algebras

  • Received: 27 October 2019 Accepted: 25 February 2020 Published: 05 March 2020
  • MSC : Primary: 06F35, 03G25; Secondary: 03G25

  • In this paper, we investigate an equivalent condition for a branchwise strongly solid gBCH-algebra to be branchwise commutative. Moreover, we show that a branchwise strongly solid gBCH-algebra is both branchwise commutative and branchwise positive implicative if and only if it is branchwise implicative.

    Citation: Muhammad Anwar Chaudhry, Asfand Fahad, Yongsheng Rao, Muhammad Imran Qureshi, Salma Gulzar. Branchwise solid generalized BCH-algebras[J]. AIMS Mathematics, 2020, 5(3): 2424-2432. doi: 10.3934/math.2020160

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  • In this paper, we investigate an equivalent condition for a branchwise strongly solid gBCH-algebra to be branchwise commutative. Moreover, we show that a branchwise strongly solid gBCH-algebra is both branchwise commutative and branchwise positive implicative if and only if it is branchwise implicative.


    Some implicational logics have contributed to give rise to the notions of a few abstract algebras such as BCK-algebras and BCI-algebras, see [1] and [2]. The recent developments in the field of artificial intelligence has contributed greatly in many aspects of daily life. The basic tools of artificial intelligence which assist in decision making are logical systems. The fundamental axioms of the implicational calculus are the motivation behind the introduction and development of BCK-algebra and BCI-algebra, see [1] and [2]. Keeping in view the strong relationships between these algebras and corresponding logics, translation procedures have been developed to relate theorems and formulas of a logic and corresponding algebra. Hence, the study of abstract algebras, which have been motivated by logical systems, and their generalization has remained a topic of interest for those who are working in the areas of artificial intelligence, logical systems and algebraic structures. Consequently, another class of algebras known as the class of BCH-algebras has been introduced in [3,4]. It has been shown [5] that the class of BCK/BCI-algebras is a proper subclass of BCH-algebras. Several aspects of this algebra have been studied in [5,6,7,8,9]. Recently, Chaudhry et al. [10] introduced the notion of a gBCH-algebra. They showed that gBCH-algebra is a generalization of BCK/BCI/BCH-algebras. Consequently, this algebra carries some connections with the BCK/BCI positive logics and corresponding systems which are used in the process of decision making in the field of artificial intelligence. The study of gBCH-algebra is a desirable topic for researchers working in the relevant areas.

    In this paper we study the class of gBCH-algebras. We define the notions of a branchwise solid gBCH-algebra and a branchwise strongly solid gBCH-algebra. We also introduce the notions of branchwise commutative, branchwise implicative and branchwise positive implicative gBCH-algebras, and we investigate relations between these classes.

    Definition 1. A BCH-algebra [3,4] is a non-empty set X with a constant 0 and a binary operation "" satisfying the following axioms:

    (Ⅰ) xx=0,

    (Ⅱ) xy=0 and yx=0 imply x=y,

    (Ⅲ) (xy)z=(xz)y

    for all x,y,zX.

    From now onward, we denote xy by xy, x(yz) by x(yz). It is well known that the class of all BCK/BCI-algebras is a proper subclass of the class of all BCH-algebras, see [5].

    Definition 2. A generalized BCH-algebra (shortly, gBCH-algebra) [10] is a non-empty set X with a constant 0 and a binary operation "" satisfying the conditions (I), (II) and the following conditions:

    (Ⅳ) (x(xy))y=0,

    (Ⅴ) (xy)x=0y

    for all x,y in X.

    Note that the condition (V) is equivalent to (xy)x=(xx)y in gBCH-algebras. Every BCH-algebra is a gBCH-algebra, since the condition (III) implies that (x(xy))y=(xy)(xy)=0 and (xy)x=(xx)y=0y for all x,yX.

    A BCH-algebra (or gBCH-algebra) is said to be proper if it does not satisfy the condition:

    (Ⅵ) ((xy)(xz))(zy)=0

    for all x,y,zX.

    Example 1. [10] Let X:={0,w,x,y,z} be a set with the following table:

    0wxyz0000yyww00yyxxx0yzyyyy00zzyyw0

    Then it is easy to see that (X,,0) is a gBCH-algebra, but not a BCH-algebra, since (xy)z=0w=(xz)y.

    This research is a continuation of Chaudhry et al. [10], and so we refer several definitions and theorems discussed in [10]. Let (X,,0) be a gBCH-algebra. We define a binary relation "" on X by xy if and only if xy=0. An element x0 is said to be a minimal element of X if xx0 implies x=x0. We denote Min(X) the set of all minimal elements of X. The set Med(X) consists of all elements xX satisfying 0(0x)=x, and we call it a medial part of X. A set B(x0):={xX|x0x}, where x0Min(X), is called a branch of X. It is known that Min(X)=Med(X) in generalized BCH-algebras (see [10]).

    Proposition 1. [10] Let (X,,0) be a gBCH-algebra. Then

    (ⅰ) 0 is a minimal element of X,

    (ⅱ) x0=x for all xX.

    Theorem 1. [10] Let (X,,0) be a gBCH-algebra. If xX, then there exists a unique x0Med(X) such that xB(x0).

    Theorem 2. [10] Let (X,,0) be a gBCH-algebra and x0,x1Med(X). Then

    (ⅰ) 0(xy)=(0x)(0y) for all x,yX,

    (ⅱ) x,yB(x0) if and only if xyB(0),

    (ⅲ) if yB(x0) and xy,yz, then x,zB(x0),

    (ⅳ) if x0x1, then B(x0)B(x1)=.

    Definition 3. A gBCH-algebra (X,,0) is said to be branchwise solid if, for any x,y,zB(a), where aMed(X),

    ((xy)(xz))(zy)=0, (3.1)

    i.e., (xy)(xz)zy.

    Note that if (X,,0), (Y,,0) are two gBCH-algebras then the cartesian product X×Y is also a gBCH-algebra with constant (0,0), where the binary operation "" on X×Y is defined by component-wise from the operations on X and Y, respectively, i.e., (x,a)(y,b):=(xy,ab) for any (x,a),(y,b)X×Y.

    Definition 4. A branchwise solid gBCH-algebra is said to be proper branchwise solid gBCH-algebra if it is not a BCH-algebra.

    Now, we give two examples of a proper branchwise solid gBCH-algebra as follows.

    Example 2. Let X be a gBCH-algebras as in Example 1 and Y:={0,p} with the binary operation defined by

    0p000pp0

    Then it is easy to see that Y×X is a branchwise solid gBCH-algebra, but not a BCH-algebra, since [(0,x)(0,y)](0,z)=(0,0)(0,w)=[(0,x)(0,z)](0,y). Hence it is a proper branchwise solid gBCH-algebra.

    Example 3. Let (X,,0) be a gBCH-algebras as in Example 1. Then (X×X,,(0,0)) is a proper gBCH-algebra, since [(x,w)(y,x)](z,z)=(y,0)(z,z)=(0,y) and [(x,w)(z,z)](y,x)=(z,y)(y,x)=(w,y). Moreover, [[(w,w)(z,z)][(x,x)(y,y)]][(y,y)(z,z)]=[(z,z)(y,y)][(y,y)(z,z)]=(w,w)(0,0)=(w,w)0. It shows that (X×X,,(0,0)) does not satisfy the condition (1) in Definition 3. However, routine calculations give that every branch B(a), aMed(X), satisfies the condition (1), hence it is a proper branchwise solid gBCH-algebra.

    Theorem 3. Let (X,,0) be a branchwise solid gBCH-algebra and let x,y,zB(a) where aMed(X). Then

    (i) if xy, then xzyz,

    (ii) if xy, then zyzx,

    (iii) if xy, yz, then xz,

    (iv) xy=x(x(xy)),

    (v) if xy, then z(zx)y.

    Proof. (ⅰ) Let xy. By Proposition 1 (ii), we have x0=x for all xX. It follows from (1) of Definition 3 that (xz)(yz)=((xz)0)(yz)=((xz)(xy))(yz)=0. Hence xzyz.

    (ⅱ) If xy, then xy=0 and hence (zy)(zx)=((zy)(zx))0=((zy)(zx))(xy)=0 by Proposition 1 (ii). Hence zyzx.

    (ⅲ) Let xy, yz. Then xy=0 and yz=0. By Proposition 1 (ii) and (1), we obtain xz=(xz)0=(xz)(xy)=((xz)(xy))0=((xz)(xy))(yz)=0, proving that xz.

    (ⅳ) Given x,yB(a). Since x(xy)y, so by Theorem 2 (iii), we obtain x(xy)B(a). Since X is a branchwise solid gBCH-algebra, so by Definition 3 and (IV) we have

    (xy)(x(x(xy)))(x(xy))y=0.

    On the other hand, (IV) gives (x(x(xy)))(xy)=0. So by (II), we have x(x(xy))=xy.

    (ⅴ) Let xy. Since z(zx)x and xB(a), by Theorem 2 (iii), we have z(zx)B(a) and z(zx)xy. By Theorem 3.1 (iii), we obtain z(zx)y.

    Definition 5. A branchwise solid gBCH-algebra (X,,0) is said to be branchwise strongly solid if, for any x,y,zB(a), where aMed(X),

    (xy)z=(xz)y. (4.1)

    To demonstrate the significance of this notion, we provide some examples. It is easy to see that Examples 1, 2 and 3 are branchwise strongly solid gBCH-algebras.

    Example 4. Let Z:={0,a,b,c,d} be a set with the following table:

    0abcd00000daa00adbbb00dcccc0dddddd0

    Then (Z,,0) is a gBCH-algebra. Let (X,,0) be a gBCH-algebra as in Example 1. Then (Z×X,,(0,0)) is a proper gBCH-algebra, where is defined component wise as described earlier. But, it is is not branchwise solid, since [[(a,0)(c,w)][(a,0)(b,w)]][(b,w)(c,w)]=[(a,0)(0,0)](0,0)=(a,0)(0,0)=(a,0)(0,0).

    Definition 6. A gBCH-algebra (X,,0) is said to be branchwise commutative if, for any x,yB(a), aMed(X), x(xy)=y(yx).

    The following theorem provides an equivalent condition for a branchwise strongly solid gBCH-algebra (X,,0) to be branchwise commutative.

    Theorem 4. Let (X,,0) be a branchwise strongly solid gBCH-algebra. Then it is branchwise commutative if and only if y(yx)=x(x(y(yx))), for all x,yB(a), aMed(X).

    Proof. Assume (X,,0) is branchwise commutative. Given x,yB(a),aMed(X), since y(yx)x, by Theorem 1 and Theorem 2 (iii), there exists uniquely x0Min(X) such that y(yx),xB(x0). By Theorem 2 (iv), we can show that a=x0. Since (X,,0) is branchwise commutative, by Proposition 1 (ii), we obtain

    x(x(y(yx)))=(y(yx))[(y(yx))x]=(y(yx))0=y(yx).

    Conversely, assume y(yx)=x(x(y(yx))), for all x,yB(a), for all aMed(X). By Theorem 2 (ii), we obtain yxB(0). It follows that 0(yx)=0. Since (X,,0) is a branchwise strongly solid gBCH-algebra, we have (y(yx))y=(yy)(yx)=0(yx)=0. Since x(xy)y, both x(xy) and y belong to B(a). Since (X,,0) is a branchwise strongly solid gBCH-algebra, by applying Theorem 3.1 (iv), we obtain

    [x(x(y(yx)))](x(xy))=[x(x(xy))][x(y(yx))]=(xy)[x(y(yx))]. (4.2)

    Since (X,,0) is a branchwise solid gBCH-algebra, by applying (4.2), we obtain

    [y(yx)][x(xy)]=[x(x(y(yx)))](x(xy))=(xy)[x(y(yx))](y(yx))y=0.

    This proves that y(yx)x(xy). If we interchange the role of x and y, we obtain x(xy)y(yx), proving that (X,,0) is branchwise commutative.

    Now, we introduce the notions of a branchwise implicative gBCH-algebra as well as of a branchwise positive implicative gBCH-algebra.

    Definition 7. A gBCH-algebra (X,,0) is said to be branchwise implicative if, for any x,yB(a), aMed(X), x(yx)=x.

    Definition 8. A gBCH-algebra (X,,0) is said to be branchwise positive implicative if xy=(xy)(y(0(0y))), for any x,yB(a), aMed(X).

    Theorem 5. Let (X,,0) be a branchwise strongly solid gBCH-algebra. If X is branchwise implicative, then it is branchwise commutative.

    Proof. Let x,yB(a),aMed(X). Since x(xy)y, by Theorem 2 (iii), we have x(xy)B(a). Moreover, since y(y(x(xy)))x(xy), we have y(y(x(xy)))B(a). Since (X,,0) is branchwise implicative, we obtain

    x(xy)=[x(xy)][y(x(xy))]. (4.3)

    for any x,yB(a),aMed(X). Since (X,,0) is branchwise strongly solid, we have

    [[x(xy)][y(x(xy))]][y(y(x(xy)))]=[[x(xy)][y(y(x(xy)))]][y(x(xy))]. (4.4)

    Since x(xy)y, by Theorem 3.1 (i) and (IV), we obtain

    [x(xy)][y(y(x(xy)))]y[y(y(x(xy)))]y(x(xy)).

    It follows that

    [[x(xy)][y(y(x(xy)))]][y(x(xy))]=0. (4.5)

    By (4.4) and (4.5), we obtain [[x(xy)][y(x(xy))]][y(y(x(xy)))]=0, i.e.,

    [x(xy)][y(x(xy))]y(y(x(xy))). (4.6)

    By (4.3) and (4.6), we obtain x(xy)y(y(x(xy))). By (Ⅳ), we have y(y(x(xy)))x(xy). This proves that x(xy)=y(y(x(xy))). By applying Theorem 4, (X,,0) is branchwise commutative.

    Theorem 6. Let (X,,0) be a branchwise strongly solid gBCH-algebra. If X is branchwise implicative, then it is branchwise positive implicative.

    Proof. Given x,yB(a),aMed(X), since X is branchwise strongly solid, by applying Theorem 2 (i) and Theorem 3.1 (iv), we obtain

    [(xy)(y(0(0y)))](xy)=[(xy)(xy)][y(0(0y))]=0(y(0(0y)))=(0y)(0(0(0y)))=(0y)(0y)=0.

    It follows that

    (xy)(y(0(0y)))xy (4.7)

    for any x,yB(a),aMed(X). Now, since 0(0y)y and yB(a), we have 0(0y)B(a). Since X is branchwise strongly solid, we obtain

    (y(xy))(0(0y))=[y(0(0y))](xy) (4.8)

    By Theorem 5, X is also branchwise commutative. It follows that

    (xy)[(xy)(y(0(0y)))]=[y(0(0y))][[y(0(0y))](xy)]=[y(0(0y))][(y(xy))(0(0y))][(4.8)]=[y(0(0y))][y(0(0y))][X:branch.imp.]=0.

    Hence we obtain xy(xy)(y(0(0y))). By combining it with (4.7) and using (II), we get xy=(xy)(y(0(0y))).

    The following theorem yields sufficient conditions for a branchwise strongly solid gBCH-algebra (X,,0) to be branchwise implicative.

    Theorem 7. Let (X,,0) be a branchwise strongly solid, branchwise commutative and branchwise positive implicative gBCH-algebra. Then it is branchwise implicative.

    Proof. Given x,yB(a),aMed(X), by Theorem 2 (ii), we obtain xy,yxB(0), i.e., 0(xy)=0(yx)=0. Since X is branchwise strongly solid, we have

    (x(yx))x=(xx)(yx)=0(yx)=0.

    This shows that x(yx)x. Since xB(a), by applying Theorem 2 (iii), we obtain x(yx)B(a). If we take x0:=0(0x), then x0x and x0Min(X)=Med(X). Since xB(a), by Theorem 2 (iii), we have x0B(a). Since x0,aMed(X), by Theorem 2 (iv), we obtain a=x0. Hence

    (0(0x))(x(yx))=x0(x(yx))=a(x(yx))=0.

    Since X is branchwise commutative, by applying Theorem 4.1, we obtain

    x(x(yx))=(yx)[(yx)(x(x(yx)))]=[(yx)(x(0(0x)))][(yx)(x(x(yx)))][X:branch.pos.imp.][x(x(yx))][x(0(0x))][Definition5](0(0x))(x(yx))[Definition5]=0.

    It follows that xx(yx). Thus we proved that x=x(yx).

    The following is a consequence of the above three theorems.

    Corollary 1. A branchwise strongly solid gBCH-algebra (X,,0) is both branchwise commutative and branchwise positive implicative if and only if it is branchwise implicative.

    The theory of a gBCH-algebra is one of recent topics in the field of algebraic structures, which has attraction to many mathematicians and computer scientists. In this article, several notions such as branchwise solid gBCH-algebras, branchwise strongly solid gBCH-algebras, branchwise commutativity, branchwise implicativity and branchwise positive implicativity have been studied. Moreover, we investigated necessary and sufficient conditions for a branchwise strongly solid gBCH-algebra to be branchwise commutative. We also developed some relationships among the branchwise implicativity and the branchwise commutativity and the branchwise positive implicativity. The sufficient condition for a gBCH-algebra to be branchwise implicative has also been proved. The notion of a gBCH-algebra provides some possibility to open the doors of BCH-algebras into the area of gBCH-algebras. The areas of the categorical aspects, graph algebras, ideals and filters in gBCH-algebras will be discussed in sequel. It will also be interesting to investigate: (i) which parts of the Theorem 3 can be proved for a gBCH-algebra and whether the condition that x,y,z are from the same branch B(a) is necessary or it may be relaxed for some parts? and (ii) an example of a branchwise solid gBCH-algebra which is not branchwise strongly solid gBCH-algebra.

    All the authors are thankful to their respective institutions for providing excellent research facilities. Moreover, the second and fourth authors are also thankful to Higher Education Commission of Pakistan. The research of the third author was supported by the Guangzhou Academician and Expert Workstation (No. 20200115-9).

    The authors hereby declare that there are no conflicts of interest regarding the publication of this paper.



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