By
Assist.prof. Hussein Hadi Abbass
University of Kufa\College of Education for girls\
Department of Mathematics
Msc_hussien@yahoo.com
Hasan Mohammed Ali Saeed
University of Kufa\ College of Mathematics and
Computer Sciences \Department of Mathematics
Has_moh2005@yahoo.com
Historical
Introduction
1966
1983
• BCK-algebra(Bose, chaudhury, K. Iseki)
• BCI-algebra (K. Iseki)
• Ideal of BCK-algebra (K. Iseki)
• BCH-algebra (Q.P.Hu, X.Li)
• Ideal of BCI-algebra (C. S. Hoo)
1991
• Ideal of a BCH-algebra (M. A. Chaudhry)
1996
2010
2011
• Closed ideal of BCH-algebra (M. A.
Chaudhry) and (H. Fakhar-ud-din)
• Fantastic ideal (A. B. Saeid )
• closed ideal with respect to an element of a BCHalgebra (H. H. Abbass, H.M.A. Saeed)
• Closed BCH-algebra with respect to an element (H.
H. Abbass, H.M.A. Saeed)
‫أن أغلب األجهزة الرقمية في أنظمة االتصاالت تقوم بنق ل المعلومات على‬
‫شكل ذبذبات مرمزة بشكل ثنائي أي ان المعلومة (‪)message‬تكون‬
‫بصيغة ‪1,0‬لذلك سوف يكون موضوع الندوة حول الشفرات الثنائية وهذه‬
‫الشفرات يمكن الحصول عليها من عناصر حق ل غالوا ) ‪ GF(2n‬وعلى سبيل‬
‫بالرمز‬
‫لها‬
‫يرمز‬
‫ادناه‬
‫المخطط‬
‫في‬
‫الذبذبة‬
‫المثال‬
‫‪10110‬‬
‫) ‪GF(2n‬‬
‫ينشأ حقل غالوا) ‪ GF(2n‬بأخذ متعددة حدود ) ‪ p(x‬غير قابلة للتحليل (بدائية) ‪ primitive‬ذات‬
‫درجة ‪ n‬تنتمي الى حلقة متعددة‬
‫‪‬‬
‫الحدود ‪Z 2 x ‬‬
‫فيتم انشاء حقل غالوا باستخدامها‬
‫‪‬‬
‫‪GF(2n )  Z2 ( x) /  p( x)  ao  a1  ... an1n1 : ao , a1,...,an1  Z2‬‬
‫حيث أن ‪ p( )  0‬اي ان ‪ ‬تكون جذراً لـــ ) ‪ p(x‬أن‬
‫) ‪GF(2n‬‬
‫وهو حقل توسيع للحقل ‪ Z 2‬ويعتبر هذا الحقل مولد لـ‬
‫مــثال ‪ :‬أنشئ حقل غالوا‬
‫) ‪GF(23‬‬
‫‪1‬‬
‫‪n‬‬
‫يحتوي على ‪ 2n‬من العناصر‬
‫‪ 2‬من الذبذبات (الشفرات الرقمية )‬
‫باستخدام متعددة ‪‬‬
‫الحدود‪p( x)  1  x  x  Z2 X‬‬
‫‪3‬‬
‫‪GF(23 )  a0  a  a22 : a0 , a1 , a2  Z 2 ‬‬
‫وبماأن‬
‫‪Z 2  0,1‬‬
‫اذن يمكن ايجاد عناصر غالوا من خالل التعويض عن‬
‫‪a0 , a1 , a2‬‬
‫ب ‪ 0,1‬فيكون هناك ‪ 2‬من األحتماالت التي تمثل عناصر ح ق ل غالوا‬
‫‪3‬‬
‫) ‪GF(23‬‬
‫من وجهة نظر الرياضيات والذي يمثل مولد لشفرات رقمية ثنائية لمصممي‬
‫األجهزة األلكترونية‬
‫‪3‬‬
‫‪2‬‬
‫‪GF(2 )  a0  a  a2 : a0 , a1 , a2  Z2 ‬‬
‫‪a1‬‬
‫‪a1‬‬
‫‪a0‬‬
‫‪0‬‬
‫‪0‬‬
‫‪0‬‬
‫‪0‬‬
‫‪0‬‬
‫‪1‬‬
‫‪0‬‬
‫‪1‬‬
‫‪0‬‬
‫‪1‬‬
‫‪0‬‬
‫‪0‬‬
‫‪1‬‬
‫‪0‬‬
‫‪1‬‬
‫‪1‬‬
‫‪1‬‬
‫‪0‬‬
‫‪1‬‬
‫‪1‬‬
‫‪0‬‬
‫‪1‬‬
‫‪1‬‬
‫‪1‬‬
‫أي أن ) ‪ GF(23‬هو مولد رسائل رقمية ثنائية‬
‫‪0‬‬
‫‪1‬‬
‫‪‬‬
‫‪2‬‬
‫‪1 ‬‬
‫‪1  2‬‬
‫‪  2‬‬
‫‪1    2‬‬
‫الشكل األتي يبين مولد شفرات رقمية ‪shift‬‬
‫‪ regester‬الذي يمثل الحق ل ) ‪ GF(24‬المولد‬
‫بواسطة متعددة الحدود‬
‫‪1000‬‬
‫‪1110‬‬
‫‪1001‬‬
‫‪0001‬‬
‫‪1101‬‬
‫‪0010‬‬
‫‪0111‬‬
‫‪0011‬‬
‫‪1010‬‬
‫‪0100‬‬
‫‪0101‬‬
‫‪1111‬‬
‫‪0110‬‬
‫‪1011‬‬
‫‪1100‬‬
‫خالل نق ل المعلومات تكون معرضة للتشويش أي بمعنى اخر وجود اشارات دخيلة التي‬
‫ممكن أن تغير المعلومات المنقولة ‪.‬لذلك ظهرت الحاجة الى مايسمى الشفرات‬
‫الكاشفة والمصححة لألخطاء‪.‬‬
‫‪0010101010‬‬
‫التشويش‬
‫التشويش بعد حين‬
‫صورة مستلمة من احدى االقمار الصناعية قبل تصحيحها‬
‫وعلى سبيل المثال تقول وكالة ناسا للفضاء األمريكية أنها أرسلت ‪ 1969‬مركبتان فضائيتان‬
‫(‪ )Mariners 6,7‬وأن هاتان المركبتان أرسلتا أكثر من ‪ 200‬صورة لكوكب المريخ كل‬
‫صورة تحتاج الى ( ‪ (5‬ماليين بت وأن معدل األرسال هو ‪ 16200‬بت في الثانية ترسل الى‬
‫األرض وأنها قد استخدمت الشفرات الكاشفة والمصححة لألخطاء‬
‫الصورة التي تم عرضها سابق ا(قبل التصحيح)‬
‫( بعد التصحيح )‬
‫نعرض الصورة المشوشة التي تم عرضها سابق اً وتظهر الصورة الحقيقية على اليسار‬
www.NASA.gov
www.NASA.gov
1.PRELIMINARIES
Definition ( 1.1 ) [5, 6] :
A BCI-algebra is an algebra (X,*,0), where X is
nonempty set, * is a binary operation and o is a
constant , satisfying the following axioms:
 for all x,y, z X:
 ((x *y) * (x * z)) * (z * y) = 0,
 (x * (x * y)) * y = 0,
 x * x = 0,
 x * y =0 and y * x = 0 imply x = y,
Definition ( 1.2 ) [11] :

A BCK-algebra is a BCI-algebra satisfying the
axiom: 0 * x = 0 for all x  X.
Definition ( 1.3 ) [10] :
A BCH-algebra is an algebra (X,*,0), where X is
nonempty set, * is a binary operation and 0 is a
constant , satisfying the following axioms:
for all x,y, z X:
x * x = 0,
x * y =0 and y * x = 0 imply x = y,
(x*y)*z=(x*z)*y
Definition ( 1.4 ) [3, 11, 12] :
In any BCH/BCI/BCK-algebra X, a partial order  is
defined by putting xy if and only if x*y = 0.
Proposition ( 1.5 ) [4, 8, 9] :
In a BCH-algebra X, the following holds for all x, y,
z  X,
x*0=x,
(x * (x * y)) * y = 0,
0 * (x * y) = (0 * x) * (0 * y) ,
0 * (0 * (0 * x)) = 0 * x ,
x  y implies 0 * x = 0 * y.
Definition ( 1.7 ) [2] :
A BCH-algebra X that satisfying in condition if 0 *
x = 0 then x = 0, for all xX is called a Psemisimple BCH-algebra.
Definition ( 1.8 ) [7] :
A BCH-algebra X is called medial if x * ( x * y ) = y ,
for all x , y  X.
Definition ( 1.9 ) [2] :
A BCH-algebra X is called an associative BCHalgebra if:
( x * y ) * z = x * ( y * z ) , for all x , y , z  X.
TYPE OF BCH-ALGEBRA
BCH-ALGEBRA
Psemisimple
Medial
associative
Definition ( 1.10 ) [9] :
Let X be a BCH-algebra . Then the set X+ = { x  X : 0
* x = 0 } is called the BCA-part of X .
Definition ( 1.12 ) [9] :
Let X be a BCH-algebra . Then the set med(X) = {x 
X : 0 * (0 * x) = x} is called the medial part of X .
PART OF BCH-ALGEBRA
BCH-algebra
BCA-part
Medial part
Theorem ( 1.14 ) [7] :
Let X be a BCH-algebra . Then x  med(X) if and
only if x*y = 0*(y*x) for all x,yX.
Definition ( 1.15 ) [9] :
Let S be a subset of a BCH-algebra X. S is called a
subalgebra of X if x*y  S for every x,yS.
Definition ( 1.16 ) [2, 7]:
Let I be a nonempty subset of a BCH-algebra X.
Then I is called an ideal of X if it satisfies:
(i) 0I
(ii) x*yI and y I imply xI.
Definition ( 1.17) [7] :
An ideal I of a BCH-algebra X is called a closed
ideal of X if 0*xI for all xI.
Definition ( 1.18 ) [7] :
An ideal I of a BCH-algebra X is called a quasiassociative ideal if 0*(0*x)=0*x, for each x  I.
Definition ( 1.19 ) [1 ] :
Let I be an ideal of X. Then I is called a fantastic
ideal of X if (x*y)*zI and zI, then
x*(y*(y*x))I, for all x, y, zX.
Definition ( 1.19 ) [1 ] :
Let I be an ideal of X. Then I is called a fantastic
ideal of X if (x*y)*zI and zI, then
x*(y*(y*x))I, for all x, y, zX.
Proposition ( 1.20 ) [1] :
If X is an associative BCI-algebra, then every ideal
is a fantastic ideal of X.
2.THE MAIN RESULTS
Definition ( 2.1 ):
Let X be a BCH-algebra and I be an ideal of X .
Then I is called a closed ideal with respect to an
element aX (denoted a-closed ideal) if a * ( 0 * x )
 I , for all x  I.
Remark ( 2.2 ):
In a BCH-algebra X , the ideal I = {0} is the closed
ideal with respect to 0 . Also , the ideal I = X is the
closed ideal with respect to all elements of X.
 Example ( 2.3 ):
Let X = {0 ,a ,b ,c }. The following table shows the
BCH-algebra on X .
*
0
a
b
c
0
0
a
b
c
a
a
0
c
b
b
b
c
0
a
c
c
b
a
0
Then I={0,a} is closed ideal with respect to( 0 and a)
,Since
0I,
If x*yI ,and yI implies xI
So I is ideal.
0*(0*0) = 0  I and 0*(0*a) = a I I is 0-closed
ideal. Also
a*(0*0)=a I and a*(0*a) = 0 I I is a-closed ideal.
Definition ( 2.4 ):
Let X be a BCH-algebra and aX. Then X is called
closed BCH-algebra with respect to a, or a-closed
BCH-algebra, if and only if every proper ideal is
closed ideal with respect to a.
Example ( 2.5 ):
Let X={0, 1, 2, 3, 4, 5}. The following table shows the
BCH-algebra structure on X.
*
0
1
2
3
4
5
0
0
0
0
0
4
4
1
1
0
0
1
4
4
2
2
2
0
2
5
4
3
3
3
3
0
4
4
4
4
4
4
4
0
0
5
5
5
4
5
2
0
I1={0, 1} , I2={0, 1, 2} and I3={0, 1, 2, 3} are all the
proper ideals [since if x*yIi and yIi  xIi ,
i=1, 2, 3 ] .
I1 is closed ideal with respect to 1 [since 1*(0*x)I1
,x I1 ]
I2 is closed ideal with respect to 1 [since 1*(0*x)I2
,x I2 ]
I3 is closed ideal with respect to 1 [since 1*(0*x)I3
,x I3 ]
Therefore,
X is 1-closed BCH-algebra.
But X is not 2-closed BCH-algebra, since
I1 is not 2-closed ideal[Since(2*(0*1))=2I1
Theorem(2.6) :
Let X is a BCH-algebra . If X=X+ (where X+ is a BCApart), then X is 0-closed BCH-algebra .
Corollary(2.7) :
Every BCK-algebra is 0-closed BCH-algebra.
BCK-algebra
BCH and BCA-algebra
0-closed BCH-algebra
Theorem(2.8) :
Let X be a 0-closed BCH-algebra. Then every
quasi-associative ideal is closed ideal.
Theorem(2.9) :
Let X be a 0-closed BCH-algebra. Then every quasiassociative ideal is subalgebra.
In 0-closed BCH-algebra
Quasi-associative
ideal
subalgebra
Closed ideal
 proposition(2.10) :
 Let X be an a-closed BCI-algebra, with aX. If X is an
associative BCI-algebra, then every a-closed ideal is a
fantastic ideal of X .