In this paper, we introduce a new notion that we call a sub branch of an element (a) related to an element (b), denoted by S(a)b in a BCC-algebra and gBCK-algebra, and we link this notion with another  notions of BCC- algebra and gBCK-algebra. We give some properties of S(a)b in a (BCC, gBCK)-algebras. Keywords: Sub branch of element related to an element, BCC-algebra, gBCK-algebra

Abbass, Husein Hadi

Abdulhussein, Mohsin shaalan

Dahham, Hayder Abbas

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.3, 2016  129  On a Sub Branch of Element Related to an Element in a (BCC, gBCK)-algebras  Assist.prof. Husein Hadi Abbass Faculty  of Education for Girls, University of Kufa  Assist. Lec. Hayder Abbas Dahham Faculty of Dentistry, University of Kufa  Assist. Lec. Mohsin shaalan Abdulhussein Faculty of physical planning, University of Kufa Abstract           In this paper, we introduce a new notion that we call a sub branch of an element (a) related to an element (b), denoted by S(a)b in a BCC-algebra and gBCK-algebra, and we link this notion with another  notions of BCC- algebra and gBCK-algebra. We give some properties of S(a)b in a (BCC, gBCK)-algebras.   Keywords:  Sub branch of element related to an element, BCC-algebra, gBCK-algebra.    INTRODUCTION     In 1966, Y. Imai and K. Iseki introduced two classes of abstract algebras:  BCK-algebra and BCI-algebra, the notion of BCI-algebra was a generalization of a BCK-algebra[4]. In 1984, Y. Komori, introduced the notion of  BCC-algebra[8]. In 1998, J. Hao. introduced the concept of ideals in a BCC-algebra[3]. In 2003, S. M. Hong, Y. B. Jun and M. A. Ozturk  construct a new algebra, called a generalized BCK-algebra (gBCK-algebra for short), which is a generalization of a positive implicative BCK-algebra [5]. The aim of this paper is to construct a new subset of (BCC, gBCK)-algebras, called a sub branch of element related to an element. We study the properties of this subset in (BCC, gBCK)-algebras   1.Preliminaries In this section, we review some basic definitions and notations of BCC-algebras, gBCK- algebras and some types of ideals, that we need in our work. Definition ( 1.1 ) [8] : By BCC-algebra we mean a non-empty set X, with a constant 0 and a binary operation”*” satisfying  the following axioms: for all x,y,zX, 1-  ((xy)  (zy)) * (x*z) = 0, 2- 0*x = 0, 3- x*0 = x, 4-  x*y = 0 and y*x = 0 imply x = y.     in X we can define a binary relation "≤" by  x ≤ y if and only if x *  y = 0, is called a BCC-order on X. A nonempty subset S of X is called a subalgebra of X if x * y S for all x, y  S.   Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.3, 2016  130  Example ( 1.2 )  [8]   : Let X = {0,1,2, 3,4} be a set with the following table:  *  0 1 2 3 4 0  0 0 0 0 0 1  1 0 1 0 0 2  2 2 0 0 0 3  3 3 1 0 0 4  4 3 4 3 0 Then (X; * , 0) is a BCC-algebra. proposition ( 1.3 ) [8] : In a BCC-algebra, the following hold,  for any x,y,zX, 1- x*x = 0, 2- (x*y)*x = 0, 3- x*y = 0 imply (x*z)*(y*z)=0.    4- x*y = 0 imply (z*y)*(z*x)=0.    Definition ( 1.4 ) [5] : By a generalized BCK-algebra (gBCK-algebra, for short) we mean a triplet (X, *, 0), where X is a nonempty set, * is a binary operation on X and 0 ∈  X, such that 1- x *  0 = x, 2-  x *  x = 0, 3- (x  y) *  z = (x *  z) *  y,  4- (x *  y) *  z = (x *  z) *  (y *  z). Example ( 1.5 ) [5] : Let X = {0, 1, 2, 3} be a set with the following table: *  0 1 2 3 0  0 0 0 0 1  1 0 1 1 2  2 0 0 0 3  3 0 0 0 Then X is a gBCK-algebra Proposition ( 1.6 ) [5] : Let X be a gBCK-algebra. Then 1- 0  x = 0, 2-  (x * y) * x = 0.  3- x * y = 0 implies (x * z) * (y * z) = 0. Definition ( 1.7 ) [3] :    A nonempty subset I is called an ideal of X if it satisfies: 1- 0I. 2-  x*yI and yI imply xI. Definition ( 1.8 ) [2] :    An ideal I  is called a closed ideal of X if: for every xI, we have 0*xI. Definition ( 1.9 ) [2] : An ideal   I is called a closed ideal with respect to an element bX (denoted b-closed ideal) if   b*(0*x)I , for all xI. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.3, 2016  131  Definition ( 1.10 ) [1] :    An ideal I of  X is called a  completely closed ideal of X if: for every x,yI, we have x*yI.  Definition ( 1.11 ) [1] : An ideal  I is called a completely closed ideal with respect to an element bX (denoted b-completely closed ideal) if   b*(x*y)I , for all x,yI. Definition ( 1.12) [2] :   An ideal I  satisfies the condition: x  I and a X\I imply x*a I, is called a *-ideal of X. Definition ( 1.13) [7] : A nonempty subset I of a BCC-algebra X is called a BCC-ideal of X, if 1- 0  I, 2- (x* y) * z  I and y  I imply x* z  I; x; y; z  X. Definition ( 1.14 ) [6] : Let X be a gBCK-algebra. A nonempty subset I of X is called a generalized BCK-ideal (gBCK-ideal, for short) of X if it satisfies the following conditions: 1- x ∈ X and a ∈ I imply a * x ∈ I,  2- x ∈ X and a, b ∈ I imply x *  (x * a) * b ∈ I.  2.The Main Results:      In this section, we define the  notion of a sub branch of element related to an element of (BCC, gBCK)-algebras, and link this  notion with another notions in BCC-algebra and gBCK-algebra. Definition   ( 2.1.1 ): Let X be a (BCC, gBCK)-algebra, we define a sub branch of element a  X related to an element         b  X is the set S(a)b={xX: (xa)*b=0} Now, we study this notion in BCC-algebra and gBCK-algebra. 2.1 The Main Results in BCC-algebra: To explain Definition(2.1.1) in BCC-algebra we give the following example. Example   (2.1.1): Let X  be a BCC-algebras in example (1.2). Then  S(1)2={0,1,2}, S(3)1={0,1,2,3}, S(1)0={0,1}. Proposition   ( 2.1.2 ):  Let X be a BCC-algebra. Then 1- 0 S(a)b a,bX. 2- a S(a)b a,bX. 3- S(0)0 ={0}. Proof : 1- Let a,b  X Then  (0* a)*b=0*b=0            [definition (1.1)(2)]  0 S(a)b a,bX. 2- Let a,b  X Then  (a*a)*b=0*b=0             [definition (1.1)(2)]  a S(a)b a,bX. 3- Let x  S(0)0. Then (x*0)*0=0 Now, (x*0)*0=x*0=x  x=0  S(0)0 ={0}. Remark ( 2.1.4 ): Let X be a BCC-algebra. The  ⋃ 𝑆(𝑎)𝑏b∈X = 𝑋. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.3, 2016  132  Proposition ( 2.1.5 ): Let X be a BCC-algebra. Then  1- If x S(a)0 x*z  S(a*z)0, zX. 2- If x S(a)0 z*a  S(z*x)0, zX. 3- x*y S(z*y)x*z, y,zX. Proof :  1- let zX and  x S(a)0  (x*a)*0=0  x*a=0  (x*z)*(a*z)=0 [proposition (1.3)(3)]   ((x*z)*(a*z))*0=0      x*z  S(a*z)0  2- let zX and  x S(a)0  (x*a)*0=0  x*a=0  (z*a)*(z*x)=0 [proposition  (1.1)(4)]   ((z*a)*(z*x))*0=0      z*a  S(z*x)0  3- Since ((x*y)*(z*y))*(x*z)=0 [definition (1.1)(1)]  x*y S(z*y)x*z Proposition  ( 2.1.6 ): Let X,Y be a BCC-algebras, x, a, bX such that x S(a)b, and f be a homomorphism  from X to Y. Then f(x) S(f(a))f(b)   Proof : Since  xS(a)b   (x*a)*b=0 Now, (f(x)*f(a))*f(b)= f(x*a)*f(b)= f((x*a)*b)=f(0)=0  f(x) S(f(a))f(b)   2.2 The Main Results in gBCK-algebra: We now give some properties of S(a)b in generalized BCk-algebra  Example  ( 2.2.1 ): Let X  be a gBCK-algebras in example (1.5). Then  S(2)1={0,1,2,3}, S(3)2={0,2,3}, Theorem  ( 2.2.2 ):  Let X be a gBCK-algebra, Then S(a)b  is a subalgebra a,bX. Proof : Let a,b  X x,y  S(a)b Then (x*a)*b=0. and  (y*a)*b=0 Now, ((x*y)*a)*b=((x*a)*y)*b       [definition (1.4)(3)] =((x*a)*b)*y                                   [definition (1.4)(3)] =0*y= 0                                           [proposition (1.6)(1)]  S(a)b  is a subalgebra Theorem  ( 2.2.3 ): Let X be a gBCK-algebra. Then S(a)b is an ideal a,bX. Proof : 1- Let a, bX  Then  (0* a)*b=0*b=0              [proposition (1.6)(1)]  0 S(a)b. 2- Let a,b X, x*y S(a)b and y S(a)b          Then  ((x*y)*a)*b=0 and (y*a)*b=0           Now,  ((x*y)*a)*b=0 Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.3, 2016  133  ((x*a)*(y*a))*b=0          [definition (1.4)(4)] ((x*a)*b)*((y*a)*b)=0   [definition (1.4)(4)] ((x*a)*b)* 0 =0              [ since  y S(a)b ] (x*a)*b=0  xS(a)b   S(a)b  is an ideal. Proposition  ( 2.2.4 ): Let X be a gBCK-algebra. Then S(a)b is a *-ideal a,bX. Proof : 1- S(a)b is an ideal a,bX                  [theorem(2.2.3)] 2- Let a,bX, x S(a)b and yX/I               (x*a)*b=0            Now,   ((x*y)*a)*b=((x*a)*y)*b [definition (1.4)(3)] =((x*a)*b)*y [definition (1.4)(3)] =0*y=0 [propisition (1.6)(1)] x*yS(a)b   S(a)b  is a *- ideal a,bX. Proposition  ( 2.2.5 ): Let X be a gBCK-algebra. Then 1- b S(a)b a,bX. 2- if x S(a)b  x S(b)a   3- if x S(a)b  x*c S(a)b*c  a,b,cX. Proof : 1- a,bX Then (b* a)*b=(b*b)*a  [definition (1.4)(3)] =0*b=0                           [proposition (1.6)(1)]  b S(a)b a,bX.  2- Let a,bX and  xS(a)b   (x*a)*b=0     (x*b)*a=0   [definition (1.4)(3)]              x S(b)a. 3- Let a,b,cX and  xS(a)b   (x*a)*b=0          Now,      ((x*a)*c)*(b*c)=0                          [proposition (1.6)(3)]        ((x*c)*a)*(b*c)=0                    [definition (1.4)(3)]        x*c S(a)b*c. Proposition  ( 2.2.6 ): Let X be a gBCK-algebra, If x S(a)0 Then x S(a)b   a,bX.    Proof : 1- Let a,bX and  x S(a)0  (x*a)*0=0  x*a=0  (x*b)*(a*b)=0 [proposition (1.6)(3)]  (x*a)*b=0                   [definition (1.4)(4)]          x  S(a)b.    Proposition  ( 2.2.7 ): Let X be a gBCK-algebra. Then S(a)b is a closed ideal a,bX.   Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.3, 2016  134  Proof : 1- S(a)b is an ideal   [ theorem (2.2.3)] 2- Let xS(a)b   Now, ((0*x)*a)*b=(0*a)*b=0*b=0 0*xS(a)b   S(a)b is a closed ideal.   Proposition  ( 2.2.8): Let X be a gBCK-algebra. Then S(a)b is a x-closed ideal  x S(a)b,a,bX.   Proof : 1- S(a)b is an ideal   [theorem (2.2.3)] 2- Let x,yS(a)b  (x*a)*b=0 Now, ((x*(0*y))*a)*b=((x*0)*a)*b=(x*a)*b= 0 x*(0*y)S(a)b   S(a)b is  x-closed ideal .       Proposition  ( 2.2.9 ): Let X be a gBCK-algebra. Then S(a)b is a completely closed ideal a,bX.   Proof : 1- S(a)b is an ideal   [theorem (2.2.3)] 2- Let x,yS(a)b   (x*a)*b=0, (y*a)*b=0 Now, ((x*y)*a)*b=((x*a)*y)*b         [definition (1.4)(3)] =((x*a)*b)*y                                     [definition (1.4)(3)] =0*y= 0                                             [since x S(a)b  ] x*yS(a)b                                   S(a)b  is a completely closed ideal a,bX Proposition  ( 2.2.10 ): Let X be a gBCK-algebra. Then S(a)b is a c-completely closed ideal a,bX, c S(a)b.   Proof : 1- S(a)b is an ideal   [theorem (2.2.3)] 2- Let c,x,y S(a)b   (c*a)*b=0, (x*a)*b=0, (y*a)*b=0 Now, ((c*(x*y))*a)*b = ((c*a)*(x*y))*b [definition (1.4)(3)]  = ((c*a)*b)*(x*y)                                    [definition (1.4)(3)] =0*(x*y)                                                   [since c S(a)b  ] = 0  c*(x*y) S(a)b    S(a)b  is an c- completely closed ideal a,bX, c S(a)b. Theorem  ( 2.2.11 ): Let X be a gBCK-algebra. Then S(a)b is a BCC-ideal a,bX.   Proof : 1- 0 S(a)b                         [theorem (2.2.3)(1)]  2- Let a, b, x, y, z X  , (x*y)*z S(a)b , y  S(a)b  (((x*y)*z)*a)*b=0 , (y*a)*b=0  Now, (((x*y)*z)*a)*b=0  Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.3, 2016  135    (((x*z)*y)*a)*b=0                          [definition (1.4)(3)]   (((x*z)*a)*((y*z)*a))*b=0              [definition (1.4)(4)]  (((x*z)*a)*b)*(((y*z)*a)*b)=0         [definition (1.4)(4)]   (((x*z)*a)*b) * (((y*a)*z)*b)=0      [definition (1.4)(3)]   (((x*z)*a)*b) * (((y*a)*b)*z)=0      [definition (1.4)(3)]   (((x*z)*a)*b) * (0*z)=0                   [since y  S(a)b]   (((x*z)*a)*b) * 0 = 0                        [proposition (1.6)(1)]   (((x*z)*a)*b) = 0                       x*z S(a)b    S(a)b  is a BCC- ideal a,bX Proposition  ( 2.2.12 ): Let X be a gBCK-algebra. Then S(a)b is a gBCK-ideal a,bX.   Proof : 1- Let xX and y S(a)b  (y*a)*b=0 Now,  ((y*x)*a)*b=((y*a)*(x*a))*b=((y*a)*b)*((x*a)*b)= 0*((x*a)*b)=0  y*x S(a)b 2- Let xX and y,z S(a)b  (y*a)*b=0, (z*a)*b=0 Now,  ((x*((x*y)*z))*a)*b=((x*a)*(((x*y)*z)*a))*b =((x*a)*(((x*y)*a)*(z*a)))*b =((x*a)*(((x*a)*(y*a))*(z*a)))*b =[(x*a)*b]*[(((x*a)*(y*a))*(z*a))*b] =[(x*a)*b]*[(((x*a)*(y*a))*b)*((z*a)*b)] =[(x*a)*b]*[(((x*a)*b)*(y*a)*b)*((z*a)*b)] =[(x*a)*b]*((x*a)*b)*0)*0 =((x*a)*b)*((x*a)*b)=0  x*(x*y)*z  S(a)b. S(a)b is a gBCK-ideal a,bX.    References 1- H. H. Abass and  H. A. Dahham, On Completely Closed Ideal With Respect To an Element Of a BH-Algebra, J. Kerbala university, Vol. 10, No. 3. Scientific 2012. 2- H. H. Abass and  H. M. A. Saeed, The Fuzzy Closed Ideal With Respect To an Element Of a BH-Algebra, Msc. Thesis Kufa university ,2011. 3- J. Hao, Ideal Theory of BCC-algebras, Sci. Math. Japo. 3, 373–381, (1998). 4- K. Is e´ki, On BCI-algebras, Math. Seminar Notes 8, 125-130, (1980). 5- S. M. hong, Y. B. jun and  M. A.Ozturk , generalizations of BCK-algebras, Scientiae Mathematicae Japonicae Online, Vol. 8, (2003), 549–557. 6- W. Dudeck and X. Zhang, On ideals and congruences in BCC-algebras, Czecho Math. J. 48(123)(1998), 21-29. 7- Y Imai and K. Iseki, On axiom system of propositional caiculi XIV, Proc. Japan Academy 42 (1966), 19-22. 8- Y. Komori, The class of BCC-algebras is not a variety, Math. Japon. 29 (1984), 391– 394. 

On a Sub Branch of Element Related to an Element in a (BCC, gBCK)-algebras

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On a Sub Branch of Element Related to an Element in a (BCC, gBCK)-algebras

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