We introduce a new class of BCI-algebras, namely the class of
branchwise implicative BCI-algebras.  This class contains the
class of implicative BCK-algebras, the class of weakly implicative
BCI-algebras (Chaudhry, 1990), and the class of medial BCI-algebras.
We investigate necessary and sufficient conditions for two types of BCI-algebras
to be branchwise implicative BCI-algebras

Muhammad Anwar  Chaudhry

International Journal of Mathematics and Mathematical Sciences

Muhammad Anwar Chaudhry

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On branchwise implicative BCI-algebras

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IJMMS 29:7 (2002) 417–425PII. S0161171202004970http://ijmms.hindawi.com© Hindawi Publishing Corp.ON BRANCHWISE IMPLICATIVE BCI-ALGEBRASMUHAMMAD ANWAR CHAUDHRYReceived 19 April 2000We introduce a new class of BCI-algebras, namely the class of branchwise implicative BCI-algebras. This class contains the class of implicative BCK-algebras, the class of weaklyimplicative BCI-algebras (Chaudhry, 1990), and the class of medial BCI-algebras. We inves-tigate necessary and suﬃcient conditions for two types of BCI-algebras to be branchwiseimplicative BCI-algebras.2000 Mathematics Subject Classiﬁcation: 06F35, 03G25.1. Introduction. Iséki and Tanaka [10] deﬁned implicative BCK-algebras and stud-ied their properties. Further, Iséki [7, 8] gave the notion of a BCI-algebra which is ageneralization of the concept of a BCK-algebra. Iséki [8] and Iséki and Thaheem [11]have shown that no proper class of implicative BCI-algebras exists, that is, such BCI-algebras are implicative BCK-algebras.Thus, a natural question arises whether it is possible to generalize the notion ofimplicativeness in such a way that this generalization not only gives us a proper classof BCI-algebras but also contains the class of implicative BCK-algebras. In this paper,we answer this question in yes by introducing the concept of a branchwise implicativeBCI-algebra. This proper class of BCI-algebras contains the class of implicative BCK-algebras, the class of weakly implicative BCI-algebras [1] and the class of medial BCI-algebras [4, 6].2. Preliminaries. A BCI-algebra is an algebra (X,∗,0) of type (2,0) satisfying thefollowing conditions:(x∗y)∗(x∗z)≤ z∗y, where x ≤y if and only if x∗y = 0, (2.1)x∗(x∗y)≤y, (2.2)x ≤ x, (2.3)x ≤y and y ≤ x imply x =y, (2.4)x ≤ 0 implies x = 0. (2.5)If (2.5) is replaced by 0≤ x, then the algebra is called a BCK-algebra. It is well knownthat every BCK-algebra is a BCI-algebra.In a BCI-algebra X, the following hold:(x∗y)∗z = (x∗z)∗y, (2.6)x∗0= x, (2.7)x ≤y implies x∗z ≤y∗z and z∗y ≤ z∗x, (2.8)418 MUHAMMAD ANWAR CHAUDHRY(x∗z)∗(y∗z)≤ x∗y, (2.9)x∗(x∗(x∗y))= x∗y (see [8]). (2.10)Definition 2.1 (see [9]). A subset I of a BCI-algebra X is called an ideal of X if itsatisﬁes0∈ I, x∗y ∈ I, y ∈ I imply x ∈ I. (2.11)Definition 2.2 (see [10]). If in a BCK-algebra X(x∗y)∗z = (x∗z)∗(y∗z) (2.12)holds for all x,y,z ∈X, then it is called positive implicative.Definition 2.3 (see [10]). If in a BCK-algebra Xx∗(x∗y)=y∗(y∗x) (2.13)holds for all x,y ∈X, then it is called commutative.Theorem 2.4 (see [10]). A BCK-algebra X is positive implicative if and only if itsatisfies(x∗y)= (x∗y)∗y ∀x,y ∈X. (2.14)It has been shown in [8, 11] that no proper classes of positive implicative BCI-algebras and commutative BCI-algebras exist and such BCI-algebras are BCK-algebrasof the corresponding type. That is why we generalized these notions and deﬁnedweakly positive implicative BCI-algebras [1] and branchwise commutative BCI-algebras[3] and studied some of their properties. Each class of these proper BCI-algebras con-tains the class of BCK-algebras of the corresponding type.Definition 2.5 (see [1]). A BCI-algebra X satisfying(x∗y)∗z = ((x∗z)∗z)∗(y∗z) ∀x,y,z ∈X (2.15)is called a weakly positive implicative BCI-algebra.Theorem 2.6 (see [1]). A BCI-algebra X is weakly positive implicative if and only ifx∗y = ((x∗y)∗y)∗(0∗y) ∀x,y ∈X. (2.16)A BCI-algebra satisfying (x ∗y)∗ (z∗u) = (x ∗ z)∗ (y ∗u) is called a medialBCI-algebra.Let X be a BCI-algebra and M = {x : x ∈ X and 0∗x = 0}. Then M is called itsBCK-part. If M = {0}, then X is called p-semisimple.It has been shown in [4, 5, 6, 13] that in a BCI-algebra X the following are equivalent:X is medial, x∗(x∗y)=y ∀x,y ∈X,0∗(0∗x)= x ∀x ∈X, X is p-semisimple. (2.17)We now describe the notions of branches of a BCI-algebra and branchwise commu-tative BCI-algebras deﬁned and investigated in [2, 3].ON BRANCHWISE IMPLICATIVE BCI-ALGEBRAS 419Definition 2.7 (see [3]). Let X be a BCI-algebra, then the set Med(X)= {x : x ∈Xand 0∗(0∗x)= x} is called medial part of X.Obviously, 0∈Med(X) and thus Med(X) is nonempty. In what follows the elementsof Med(X) will be denoted by x0,y0, . . . . It is known that Med(X) is a medial sub-algebra of X and for each x ∈ X, there is a unique x0 = 0∗ (0∗x) ∈ Med(X) suchthat x0 ≤ x (see [3]). Further, Med(X), in general, is not an ideal of X. Obviously, fora BCK-algebra X, Med(X)= {0} and hence is an ideal of X.Definition 2.8 (see [3]). Let X be a BCI-algebra and x0 ∈ Med(X), then the setB(x0)={x : x∈X and x0∗x=0} is called a branch ofX determined by the element x0.The following theorem (proved in [2, 3]) shows that the branches of a BCI-algebra Xare pairwise disjoint and form its partition. So the study of branches of a BCI-algebraX plays an important role in investigation of the properties of X. Obviously, a BCK-algebra X is a one-branch BCI-algebra and in this case X = B(0).Theorem 2.9 (see [2, 3]). Let X be a BCI-algebra with medial part Med(X), then(i) X =∪{B(x0) : x0 ∈Med(X)}.(ii) B(x0)∩B(y0)=φ, x0,y0 ∈Med(X), and x0 ≠y0.(iii) If x,y ∈ B(x0), then 0∗x = 0∗y = 0∗x0 = 0∗y0 and x∗y,y∗x ∈M .Definition 2.10 (see [3]). A BCI-algebra X is said to be branchwise commutativeif and only if for x0 ∈Med(X), x,y ∈ B(x0), the following equality holds:x∗(x∗y)=y∗(y∗x). (2.18)Since a BCK-algebra is a one-branch BCI-algebra, therefore, it is commutative if andonly if it is branchwise commutative.Theorem 2.11 (see [3]). A BCI-algebra X is branchwise commutative if and only ifx∗(x∗y)=y∗(y∗(x∗(x∗y))) ∀x,y ∈X. (2.19)3. Branchwise implicative BCI-algebras. In this section, we deﬁne branchwise im-plicative BCI-algebras. We show that this proper class of BCI-algebras contains theclass of implicative BCK-algebras [10], the class of weakly implicative BCI-algebras [1]and the class of medial BCI-algebras. We also ﬁnd necessary and suﬃcient conditionsfor two types of BCI-algebras to be branchwise implicative.Definition 3.1 (see [10]). A BCK-algebra X is said to be implicative if and only ifx∗(y∗x)= x ∀x,y ∈X. (3.1)It has been shown in [8, 11] that no proper class of implicative BCI-algebras exists.Due to this reason we generalized the notion of implicativeness to weak implicative-ness [1] mentioned below.Definition 3.2 (see [1]). A BCI-algebra X is said to be weakly implicative if andonly ifx = (x∗(y∗x))∗(0∗(y∗x)) ∀x,y ∈X. (3.2)420 MUHAMMAD ANWAR CHAUDHRYWe further generalize this concept and ﬁnd a generalization of the following well-known result of Iséki [10].Theorem 3.3. An implicative BCK-algebra is a positive implicative and commuta-tive BCK-algebra.Definition 3.4. A BCI-algebra X is said to be a branchwise implicative BCI-algebraif and only ifx∗(y∗x)= x ∀x,y ∈ B(x0)and x0 ∈Med(X). (3.3)Example 3.5. Let X = {0,1,2,} in which ∗ is deﬁned by∗ 0 1 20 0 0 21 1 0 22 2 2 0  0 21Then X is a branchwise implicative BCI-algebra. This shows that proper branchwiseimplicative BCI-algebras exist.Remark 3.6. (i) Since a BCK-algebra is a one-branch BCI-algebra, therefore, it isimplicative if and only if it is branchwise implicative.(ii) Let X be weakly implicative and let x,y ∈ B(x0), x0 ∈ Med(X), then usingTheorem 2.9(iii), we get y ∗ x ∈ M . Thus 0∗ (y ∗ x) = 0. So x = (x ∗ (y ∗ x))∗(0∗(y∗x)) reduces to x = x∗(y∗x). Hence every weakly implicative BCI-algebra isbranchwise implicative BCI-algebra. But the branchwise implicative BCI-algebra X ofExample 3.5 is not weakly implicative because (1∗ (2∗1))∗ (0∗ (2∗1)) = (1∗2)∗(0∗2)= 2∗2= 0≠ 1.(iii) It is known that each branch of a medial BCI-algebra X is a singleton. Let Xbe a medial BCI-algebra and x0 ∈Med(X). Then B(x0)= {x0}. Hence x0∗(x0∗x0)=x0∗0= x0, which implies that X is branchwise implicative.Thus the class of branchwise implicative BCI-algebras contains the class of implica-tive BCK-algebras, the class of weakly implicative BCI-algebras, and the class of medialBCI-algebras. We now prove the following results.Lemma 3.7. Let X be a BCI-algebra. If x,y ∈ X and x ≤ y , then x,y ∈ B(x0) forx0 ∈Med(X).Proof. Let x ∈ X, then there is a unique x0 = 0∗ (0∗x) ∈ Med(X) such thatx ∈ B(x0). Now x0 ∗y = (0∗ (0∗x))∗y = (0∗y)∗ (0∗x) ≤ x ∗y = 0. Hencex0∗y = 0, which implies y ∈ B(x0).Theorem 3.8. If X is a branchwise implicative BCI-algebra, then it is branchwisecommutative.Proof. Let x,y ∈ X, then x∗ (x∗y) ≤ y and Lemma 3.7 imply that x∗ (x∗y)and y ∈ B(y0) for some y0 ∈ Med(X). Since X is branchwise implicative, thereforeON BRANCHWISE IMPLICATIVE BCI-ALGEBRAS 421using (3.3), we get(x∗(x∗y))∗(y∗(x∗(x∗y)))= x∗(x∗y). (3.4)Using (2.2) and (2.8), we getx∗(x∗y)= (x∗(x∗y))∗(y∗(x∗(x∗y)))≤y∗(y∗(x∗(x∗y)))≤ x∗(x∗y). (3.5)Thusx∗(x∗y)=y∗(y∗(x∗(x∗y))), (3.6)which along with Theorem 2.11 implies that X is branchwise commutative.Theorem 3.9. If X is a branchwise implicative BCI-algebra, then it satisfies(x∗y)∗(0∗y)= (((x∗y)∗y)∗(0∗y))∗(0∗y). (3.7)Proof. Since X is branchwise implicative, therefore Theorem 3.8 implies that Xis branchwise commutative. Let x,y ∈ X. Since (x ∗ y) ∗ (0 ∗ y) ≤ x, thereforeLemma 3.7 implies that (x∗y)∗(0∗y), x ∈ B(x0). Now branchwise implicativenessof X implies((x∗y)∗(0∗y))∗(x∗((x∗y)∗(0∗y)))= (x∗y)∗(0∗y), (3.8)which, using (2.6) twice, gives((x∗(x∗((x∗y)∗(0∗y))))∗y)∗(0∗y)= (x∗y)∗(0∗y). (3.9)Using branchwise commutativeness of X, from (3.9) we get((((x∗y)∗(0∗y))∗(((x∗y)∗(0∗y))∗x))∗y)∗(0∗y)=(x∗y)∗(0∗y), (3.10)which implies(((x∗y)∗(0∗y))∗y)∗(0∗y)= (x∗y)∗(0∗y), (3.11)so(((x∗y)∗y)∗(0∗y))∗(0∗y)= (x∗y)∗(0∗y). (3.12)Remark 3.10. Since a BCK-algebra is a one-branch BCI-algebra, therefore an im-plicative BCK-algebra is commutative. Further, for a BCK-algebra 0∗y = 0 and thus(3.7) reduces to x∗y = (x∗y)∗y , which implies X is positive implicative. So we getTheorem 3.3, a well-known result of Iséki [10], as a corollary from Theorems 3.8 and3.9.We now investigate necessary and suﬃcient conditions for two types of BCI-algebrasto be branchwise implicative.422 MUHAMMAD ANWAR CHAUDHRYTheorem 3.11. A BCI-algebra X, with Med(X) as an ideal of X, is a branchwiseimplicative BCI-algebra if and only if it is branchwise commutative and satisfies(x∗y)∗(0∗y)= (((x∗y)∗y)(0∗y))∗(0∗y) ∀x,y ∈X. (3.13)Proof. (⇒) Suﬃciency follows from Theorems 3.8 and 3.9.(⇐) For necessity we consider x,y ∈ X such that x,y ∈ B(x0) for some x0 ∈Med(X). Now from Theorem 2.9(iii), we get x∗y and y ∗x ∈ M . So 0∗ (x∗y) =0∗(y∗x)= 0. Further, (x∗(y∗x))∗x = (x∗x)∗(y∗x)= 0∗(y∗x)= 0, sox∗(y∗x)≤ x. (3.14)Now (3.14) along with Lemma 3.7 implies x∗(y∗x) and x belong to the branch deter-mined by x, that is, B(x0). Hence x,y and x∗(y∗x)∈ B(x0). Since X is branchwisecommutative, therefore,(x∗(x∗(y∗x)))∗(0∗x)= [(y∗x)∗((y∗x)∗(x∗(x∗(y∗x))))]∗(0∗x)= [(y∗x)∗(0∗x)]∗[(y∗x)∗(x∗(x∗(y∗x)))] (using (2.6))=[(((y∗x)∗x)∗(0∗x))∗(0∗x)]∗[(y∗x)∗(x∗(x∗(y∗x)))] (using (3.13)).(3.15)Now by using (2.6) three times, we get(x∗(x∗(y∗x)))∗(0∗x)= [[[(y∗x)∗((y∗x)(x∗(x∗(y∗x))))]∗x]∗(0∗x)]∗(0∗x).(3.16)Since x,y and x∗(y∗x)∈ B(x0), therefore x∗y,y∗x,x∗(x∗(y∗x))∈M = B(0).Since X is branchwise commutative, therefore,(x∗(x∗(y∗x)))∗(0∗x)= [[[(x∗(x∗(y∗x)))∗((x∗(x∗(y∗x)))∗(y∗x))]∗x]∗(0∗x)]∗(0∗x)= ((((x∗(x∗(y∗x)))∗0)∗x)∗(0∗x))∗(0∗x)= (((x∗(x∗(y∗x)))∗x)∗(0∗x))∗(0∗x)= ((0∗(x∗(y∗x)))∗(0∗x))∗(0∗x)= (((0∗x)∗(0∗(y∗x)))∗(0∗x))∗(0∗x)= (((0∗x)∗(0∗x))∗(0∗(y∗x)))∗(0∗x)= (0∗(0∗(y∗x)))∗(0∗x)= (0∗0)∗(0∗x)= 0∗(0∗x).(3.17)Hence(x∗(x∗(y∗x)))∗(0∗x)= 0∗(0∗x)∈Med(X). (3.18)ON BRANCHWISE IMPLICATIVE BCI-ALGEBRAS 423But (2.10) implies 0∗(0∗(0∗x))= 0∗x. So 0∗x ∈Med(X). Since Med(X) is an idealof X, therefore, x∗(x∗(y∗x))∈Med(X). Hencex∗(x∗(y∗x))= 0∗(0∗(x∗(x∗(y∗x)))). (3.19)Since x∗(x∗(y∗x))∈M = B(0), therefore, 0∗(x∗(x∗(y∗x)))= 0. Thus x∗(x∗(y∗x))= 0, which givesx ≤ x∗(y∗x). (3.20)Using (3.14) and (3.20), we getx = x∗(y∗x) ∀x,y ∈ B(x0). (3.21)Hence X is branchwise implicative. This completes the proof.Remark 3.12. Since in a BCK-algebra X, Med(X) = {0} is always an ideal of X,therefore the following well-known result regarding BCK-algebra follows as a corollaryfrom Theorem 3.11.Corollary 3.13. A BCK-algebra is implicative if and only if it is positive implicativeand commutative.Remark 3.14. The following example shows that there exist proper BCI-algebras inwhich Med(X) is an ideal. Thus the condition, Med(X) is an ideal ofX, in Theorem 3.11is not unnatural.Example 3.15 (see [12, Example 2]). The set X = {0,1,2,3} with the operation ∗deﬁned as∗ 0 1 2 30 0 0 2 21 1 0 3 22 2 2 0 03 3 2 1 0   0 21 3is a proper BCI-algebra. Here Med(X)= {0,2} is an ideal of X. Further, X is branchwiseimplicative but is not medial.Definition 3.16. Let X be a BCI-algebra. Two elements x,y of X are said to becomparable if and only if either x∗y = 0 or y∗x = 0, that is, either x ≤y or y ≤ x.Definition 3.17. Let X be a BCI-algebra. If x0 ∈Med(X) and x0 ≠ 0, then B(x0),the branch of X determined by x0, is called a proper BCI-branch of X.Theorem 3.18. Let X be a BCI-algebra such that any two elements of a properBCI-branch of X are comparable. Then X is branchwise implicative if and only if X isbranchwise commutative and satisfies(x∗y)∗(0∗y)= (((x∗y)∗y)∗(0∗y))∗(0∗y) ∀x,y ∈X. (3.22)424 MUHAMMAD ANWAR CHAUDHRYProof. (⇒) Suﬃciency follows from Theorems 3.8 and 3.9.(⇐) For necessity we consider the following two cases.Case 1. Let x,y ∈ B(0) = M . Then 0∗y = 0∗x = 0 and hence (3.22) becomesx∗y = (x∗y)∗y . Further, (x∗ (y∗x))∗x = (x∗x)∗ (y ∗x) = 0∗ (y∗x) = 0.Hencex∗(y∗x)≤ x. (3.23)Since x∗y ∈M = B(0) and X is branchwise commutative, therefore,x∗(x∗(y∗x))= (y∗x)∗((y∗x)∗x)= (y∗x)∗(y∗x)= 0. (3.24)Thusx∗≤ x∗(y∗x). (3.25)From (3.23) and (3.25), we get x = x∗(y∗x) for all x,y ∈ B(0).Case 2. Let x,y ∈ B(x0), where x0 ∈ Med(X) and x0 ≠ 0. Thus x∗y ∈ M andy∗x ∈M . So 0∗(x∗y) = 0 and 0∗(y∗x) = 0. Further, taking y = x∗y in (3.22),we getx∗(x∗y)= (x∗(x∗y))∗(x∗y) ∀x,y ∈ B(x0). (3.26)Interchanging x and y in (3.26), we gety∗(y∗x)= (y∗(y∗x))∗(y∗x) ∀x,y ∈ B(x0). (3.27)Since x,y are comparable, therefore, either y∗x = 0 or x∗y = 0. If y∗x = 0, thenx∗(y∗x)= x∗0= x. (3.28)If x∗y = 0, then branchwise commutativeness of X givesy∗(y∗x)= x∗(x∗y)= x∗0= x. (3.29)Using (3.27) and (3.29), we getx = x∗(y∗x). (3.30)Thus X is branchwise implicative.Remark 3.19. The following example shows that the conditions Med(X) is an idealof X and any two elements of a proper BCI-branch of X are comparable cannot beremoved from Theorems 3.11 and 3.18, respectively.Example 3.20. Let X = {0,1,2,3,4,5} in which ∗ is deﬁned by∗ 0 1 2 3 4 50 0 0 3 2 3 31 1 0 3 2 3 32 2 2 0 3 0 03 3 3 2 0 2 24 4 2 1 3 0 15 5 2 1 3 1 0     1 4 530 2ON BRANCHWISE IMPLICATIVE BCI-ALGEBRAS 425Routine calculations give that X is a BCI-algebra, which is branchwise commutativeand satisﬁes (3.22). But we note that(1) Med(X)= {0,2,3} is not an ideal of X because 4∗3= 3∈Med(X), 3∈Med(X)but 4 	∈ Med(X). Further, X is not branchwise implicative because 4,5 ∈ B(2)and 4∗(5∗4)= 4∗1= 2≠ 4;(2) the elements 4 and 5 of B(2) are not comparable and also X is not branchwiseimplicative.Combining Theorems 3.11 and 3.18, we get the following theorem.Theorem 3.21. Let X be a BCI-algebra such that either Med(X) is an ideal of X orevery pair of elements of a proper BCI-branch of X are comparable, then X is branch-wise implicative if and only if X is branchwise commutative and satisfies (3.22).Acknowledgement. The author gratefully acknowledges the support providedby King Fahd University of Petroleum and Minerals during this research.References[1] M. A. Chaudhry, Weakly positive implicative and weakly implicative BCI-algebras, Math.Japon. 35 (1990), no. 1, 141–151.[2] , On BCH-algebras, Math. Japon. 36 (1991), no. 4, 665–676.[3] , Branchwise commutative BCI-algebras, Math. Japon. 37 (1992), no. 1, 163–170.[4] W. A. Dudek, On some BCI-algebras with the condition (S), Math. Japon. 31 (1986), no. 1,25–29.[5] C. S. Hoo, BCI-algebras with condition (S), Math. Japon. 32 (1987), no. 5, 749–756.[6] C. S. Hoo and P. V. Ramana Murty, Quasicommutative p-semisimple BCI algebras, Math.Japon. 32 (1987), no. 6, 889–894.[7] K. Iséki, An algebra related with a propositional calculus, Proc. Japan Acad. 42 (1966),26–29.[8] , On BCI-algebras, Math. Sem. Notes Kobe Univ. 8 (1980), no. 1, 125–130.[9] K. Iséki and S. Tanaka, Ideal theory of BCK-algebras, Math. Japon. 21 (1976), no. 4, 351–366.[10] ,An introduction to the theory of BCK-algebras, Math. Japon. 23 (1978), no. 1, 1–26.[11] K. Iséki and A. B. Thaheem, Note on BCI-algebras, Math. Japon. 29 (1984), no. 2, 255–258.[12] E. H. Roh and Y. Huang, J-semisimple BCI-algebras, Math. Japon. 49 (1999), no. 2, 213–216.[13] L. Tiande and X. Changchang, p-radical in BCI-algebras, Math. Japon. 30 (1985), no. 4,511–517.MuhammadAnwar Chaudhry: Department ofMathematical Sciences, King Fahd Uni-versity of Petroleum and Minerals, Dhahran 31261, Saudi ArabiaE-mail address: chaudhry@kfupm.edu.saSubmit your manuscripts athttp://www.hindawi.comHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical Problems in EngineeringHindawi Publishing Corporationhttp://www.hindawi.comDifferential EquationsInternational Journal ofVolume 2014Applied MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical PhysicsAdvances inComplex AnalysisJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014OptimizationJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Operations ResearchAdvances inJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Function SpacesAbstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal of Mathematics and Mathematical SciencesHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AlgebraDiscrete Dynamics in Nature and SocietyHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Decision SciencesAdvances inDiscrete MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Stochastic AnalysisInternational Journal of

On branchwise implicative BCI-algebras

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