المحاضرة 1 - University of Kufa
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Transcript المحاضرة 1 - University of Kufa
By
Assist.prof. Hussein Hadi Abbass
University of Kufa\College of Education for girls\
Department of Mathematics
[email protected]
Hasan Mohammed Ali Saeed
University of Kufa\ College of Mathematics and
Computer Sciences \Department of Mathematics
[email protected]
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 ... an1n1 : ao , a1,...,an1 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 a22 : 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 xy 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 xX 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,yX.
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,yS.
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) 0I
(ii) x*yI and y I imply xI.
Definition ( 1.17) [7] :
An ideal I of a BCH-algebra X is called a closed
ideal of X if 0*xI for all xI.
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)*zI and zI, then
x*(y*(y*x))I, for all x, y, zX.
Definition ( 1.19 ) [1 ] :
Let I be an ideal of X. Then I is called a fantastic
ideal of X if (x*y)*zI and zI, then
x*(y*(y*x))I, for all x, y, zX.
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 aX (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
0I,
If x*yI ,and yI implies xI
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 aX. 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*yIi and yIi xIi ,
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))=2I1
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 aX. If X is an
associative BCI-algebra, then every a-closed ideal is a
fantastic ideal of X .