1 The Closed Ideal With Respect To an Element Of

The Closed Ideal With Respect To an Element Of a BH-Algebra
BY
Assist. prof. Hussein Hadi Abbass
University of Kufa \ College of Education for girls\ Department of Mathematics
[email protected]
Assist. Lecturer Hasan Mohammed Ali Saeed
University of Kufa\College of Engineering\Department of Material Engineering
[email protected]
Abstract
In this paper, we generalize some basic concepts to a BH-algebra, we find the relation
between the notion of an associative BH-algebra and a Group of abstract algebra and
we define the notion of a closed ideal with respect to an element of a BH-algebra
which is a generalization of the notion of a closed ideal with respect to an element of
a BCH-algebra.
Keywords: BCH-algebra, BH-algebra, b-closed ideal in a BH-algebra
Introduction
In 1983, (Q. P. Hu) and (X. Li) introduced the notion of BCH-algebra which are a
generalization of BCK/BCI-algebras [7]. In 1991, (M. A. Chaudhry) introduced the
notions of BCA-part of a BCH-algebra, medial part of a BCH-algebra, subalgebra of a
BCH-algebra, ideals and closed ideals in a BCH-algebra[4]. In 1996, (M.A. Chaudhry
and H. Fakhar-Ud-Din) introduced the notions of a medial BCH-algebra[5].
In 1998, ( Y. B. Jun ) introduced the notion of a BH-algebra, which is a
generalization of BCH-algebras[10]. Then, (Y. B. Jun, E. H. Roh, H. S. Kim and Q.
Zhang) discussed more properties on BH-algebras [6, 8, 10]. In 2009, (A. B. Saeid,
A. Namdar and R.A. Borzooei) introduced the notions of a p-semisimple BCHalgebra, an associative BCH-algebra[1]. In 2011 , (H. H. Abbass) and (H. M. A.
Saeed) introduced the notion of a closed ideal with respect to an element of a BCHalgebra[3].
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In this paper, we generalize the notions of (a p-semisimple BCH-algebra, an
associative BCH-algebra, medial BCH-algebra, BCA-part of a BCH-algebra and
medial part of a BCH-algebra) about a BH-algebra and find the relation between the
notion of an associative BH-algebra an a Group of abstract algebra firstly. Then we
define the notion of a closed ideal with respect to an element of a BH-algebra which
is a generalization of the notion of a closed ideal with respect to an element of a BCHalgebra which we defined in previous paper[3]. Finally we concept the notion of a
closed ideal with respect to an element of a BH-algebra and ideals of a BCH-algebra.
1. Preliminaries
In this section we recall some basic concepts about BCH-algebra, p-semi simple BCHalgebra ,medial BCH-algebra , associative BCH-algebra, BCA-part of a BCH-algebra ,
medial part of a BCH-algebra, Ideal of a BCH-algebra, Closed Ideal of a BCH-algebra,
Closed Ideal with respect to an element of a BCH-algebra, BH-algebra, BH-subalgebra and
ideal of a BH-algebra, with some propositions and theorems.
Definition (1.1) : [7]
A BCH-algebra is an algebra (X,*,0), where X is a nonempty set, * is a binary operation
and 0 is a constant , satisfying the following axioms:
i.
x * x = 0, x X.
ii.
x * y =0 and y * x = 0 imply x = y, x, y X.
iii.
( x * y ) * z = ( x * z ) * y, x, y, z X.
Definition (1.2) : [1]
A BCH-algebra X satisfying condition 0 * x = 0
algebra
x = 0 is called a P-semisimple BCH-
Definition (1.3) : [5]
A BCH-algebra X is called medial if x * ( x * y ) = y , for all x, y
X
Definition (1.4) : [1]
A BCH-algebra X is called an associative BCH-algebra if:
( x * y ) * z = x * ( y * z ) , for all x , y , z X.
Definition (1.5) : [4]
Let X be a BCH-algebra . Then the set X+ = { x
X : 0 * x = 0 } is called the BCA-part of X.
Remark (1.6) : [4]
The BCA-part X+ of X is a nonempty since 0 *0 = 0 gives 0
X+ .
2
Definition (1.7) : [4]
Let X be a BCH-algebra . Then the set
med(X) = {x X : 0 * (0 * x) = x} is called the medial part of X .
Remark (1.8) : [4]
In a BCH-algebra X the medial part (med(X)) of X is a nonempty, since 0 * (0 *0) = 0 gives
0 med(X) .
Proposition (1.9) : [5]
Let X be a BCH-algebra. Then 0*x Med(X) for all x X, that is, 0*(0*(0*x)) = 0*x.
Theorem (1.10) : [5]
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 (1.11) : [5]:
Let I be a nonempty subset of a BH-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 (1.12) : [4]
An ideal I of a BCH-algebra X is called a closed ideal of X if: for every x I, we have
0*x I.
Definition (1.13): [3]
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 b X (denoted by b-closed ideal) if: b*(0*x) I, for all x I.
Remark (1.14): [3]
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.
Definition (1.15) : [2,10]
A BH-algebra is a nonempty set X with a constant 0 and a binary operation * satisfying the
following conditions:
i.
x * x = 0, x X.
ii.
x * y = 0 and y * x = 0 imply x = y, x, y X.
iii.
x *0 = x, x X.
Remark (1.16) : [10]
Every BCH-algebra is a BH-algebra, but the converse is not true.
3
Definition (1.17) : [10]
A nonempty subset S of a BH-algebra X is called a BH-Subalgebra or Subalgebra of X if
x * y S, for all x, y S.
Definition (1.18) : [9]:
Let I be a nonempty subset of a BH-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
2. The main results
In this section, we generalize the notions of a p-semisimple BH-algebra, medial BHalgebra, associative BH-algebra, medial part of a BH-algebra and a BCA-part of a BH-algebra
firstly. Then we define the notion of a Closed Ideal with respect to an element of a BHalgebra, finally we study some properties of this conventions.
Remark (2.1) :
We generalize the notion of a p-semisimple BCH-algebra to a BH-algebra as follows
A BH-algebra X satisfying the condition 0 * x = 0
x = 0 is called a P-semisimple BHalgebra
Example (2.2):
Consider the BH-algebra X = {0, a, b, c} with The following operation table.
*
0
a
b
0
a
b
0
a
0
c
a
b
c
0
b
c
b
a
c
Then X is a p-semisimple BH-algebra, since 0*x = 0
c
c
b
a
0
x = 0.
Remark (2.3) :
We generalize the notion of a medial BCH-algebra to a BH-algebra as follows:
A BH-algebra X is called medial if x * ( x * y ) = y , for all x, y X
Example (2.4):
Let X be the BH-algebra in example (2.2). Then X is a medial BH-algebra, since x*(x*y) =
y, x, y X
4
Remark (2.5) :
We generalize the notion of an associative BCH-algebra to a BH-algebra as follows:
A BH-algebra X is called an associative BH-algebra if: (x*y)*z = x*(y*z), x , y , z X.
Example (2.6):
Let X be the BH-algebra in example (2.2). Then X is an associative BH-algebra, since
(x*y)*z = x*(y*z), x, y, z X.
Theorem (2.7)
Every associative BH-algebra is a group
Proof
Let (X,*, 0) be an associative BH-algebra
To prove that (X, *) is a group
1. Since a*b X, a, b X X is a closed under *
2. X is an associative
[Since a*(b*c)=(a*b)*c. By definition(2.5)]
3. the identity element is 0 , Since
Let a X
i. a*0 = a
[Since x*0 = x, x X. By definition (1.15)]
ii. 0*a= (a*a)*a
[Since x*x = 0, x X. By definition(1.15)]
= a*(a*a)
[Since X is an associative BH-algebra. By definition(2.5)]
= a*0
[Since x*x = 0, x X. By definition(1.15)]
=a
[Since x*0 = x, x X. By definition(1.15)]
4. let a X a*a=0
Therefore,
(X,*) is a group.
a-1 = a
Remark (2.8)
The converse of the above theorem is not necessary to be true as in the following example
Example (2.9)
Let (Z,+) be a group of the integer number under the binary operation of an addition and
the identity element is 0. Then (Z, +, 0) is not an associative BH-algebra, Since
1 Z, but 1+1 = 2 0 (Z,+,0) is not a BH-algebra and consequently it is not an associative
BH-algebra
Remark (2.10):
If X is not associative BH-algebra. Then the theorem(2.7) is not true as in the following
example
5
Example (2.11):
Consider the BH-algebra X = {0, a, b, c, d} with The following operation table.
*
0
a
b
c
0
0
0
0
0
a
0
0
a
a
b
b
0
0
b
c
c
c
0
c
d
d
d
d
d
the set X is not a Group, because a*(b*c) = a*0 = a
d
d
d
d
d
0
(a*b)*c = 0*c = 0.
Theorem (2.12)
Every associative BH-algebra is an abelian Group
Proof
Let (X,*,0) be an associative BH-algebra
By theorem(2.7) we get
(X,*) is a Group
To prove that X is an abelian Group
Let a, b X a*b X
To prove that a*b = b*a
a*b = 0*(a*b)
[Since X be a Group and 0 be an identity 0*x = x =x*0, x X]
= (b*b)*(a*b)
[Since x*x = 0, x X. By definition(1.15)]
= b*(b*(a*b))
[Since X is an associative BH-algebra. By definition(2.5)]
= b*((b*a)*b)
[Since X is an associative BH-algebra. By definition(2.5)]
= (b*(b*a))*b
[Since X is an associative BH-algebra. By definition(2.5)]
= (b*(b*a))*(b*0)
[Since x*0 = x, x X. By definition(1.15)]
= (b*(b*a))*(b*(a*a))
[Since x*x = 0, x X. By definition(1.15)]
= (b*(b*a))*((b*a)*a)
[Since X is an associative BH-algebra. By definition(2.5)]
= b*((b*a)*((b*a))*a))
[Since X is an associative BH-algebra. By definition(2.5)]
= b*(((b*a)*(b*a))*a)
[Since X is an associative BH-algebra. By definition(2.5)]
= b*(0*a)
[Since x*x = 0, x X. By definition(1.15)]
= (b*0)*a
[Since X is an associative BH-algebra. By definition(2.5)]
= b*a
[Since x*0 = x, x X. By definition(1.15)]
Therefore, (X,*) is an abelian group.
Remark (2.16) :
We generalize the notion of a BCA-part of a BCH-algebra to a BH-algebra as follows:
Let X be a BH-algebra . Then the set X+ = { x X : 0 * x = 0 } is called the BCA-part of X .
Example (2.17):
Let X be the BH-algebra in example (2.11). Then X+ = {0, a, b, c} is the BCA-part of a
BH-algebra X, since
0*0 = 0 ,
0*a = 0 ,
0*b = 0 and
0*c = 0 . Hence each of 0, a, b, c X+
6
Remark (2.18) :
We generalize the notion of a medial part of a BCH-algebra to a BH-algebra as follows:
Let X be a BH-algebra . Then the set med(X) = {x X : 0 * (0 * x) = x} is called the medial
part of X .
Example (2.19):
Consider X be the BH-algebra in example (2.11). Then the set med(X) = {0, d} is a medial
part of a BH-algebra X, since
0*(0*0) = 0*0 = 0 0 med(X) and
0*(0*d) = 0*d = d d med(X)
Definition (2.20):
Let X be a BH-algebra and I be an ideal of X . Then 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.
Remark (2.21):
In a BH-algebra X , the ideal I = {0} is 0-closed ideal. Also , the ideal I = X is b-closed
ideal, b X.
Example (2.22):
Consider the BH-algebra X = {0, a, b, c} where * is defined as follows:
*
0
a
b
c
0
0
c
0
b
a
a
0
0
0
b
b
b
0
c
c
c
c
a
0
Then the set I ={0, a} is a-closed ideal, Since
(1) I is an ideal, because
[0*0=0 I, 0 I 0 I a*0=a I, 0 I a I b*0=b I
[0*a=c I
a*a=0 I, a I a I
b*a = b I
c*0=c I c I]
c*a = c I]
0 I
a I.
(2) [a*(0*0) = a*0 = a I
a*(0*a) = a*c = 0 I]
[ i.e. a*(0*x) I, x I] I is a-closed ideal of X.
But I is not 0-closed ideal, Since
0*(0*a) = 0*c = b
I
Proposition (2.23) :
Let X be a BH-algebra , and I be an ideal of X. If x
I ,then I is not x-closed ideal of X.
proof
let x I ,then
x*(0*0) = x*0
[Since x*x = 0, x X. By definition(1.15) of a BH-algebra]
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But x*0 = x
x*(0*0) = x I
Therefore,
I is not x-closed ideal of X.
[Since x*0 = x, x X. By definition(1.15) of a BH-algebra]
Theorem (2.24):
Let X= X+ be a BH-algebra and I be an ideal of X. Then I is a b-closed ideal, b I.
Proof
Let b I,
To prove that I is a b-closed ideal
let x I ,then we have
b*(0*x) = b*0
=b
b*(0*x) = b I
Therefore,
I is a b-closed ideal of X, b I.
[Since X=X+ 0*x = 0, x X+. By definition(2.16)]
[Since x*0 = x, x X. By definition(1.15)]
Theorem ( 2.25):
Let { Ii , i } be a family of b-closed ideals of a BH-algebra X. then
is a b-closed ideal of X,
Proof:
To prove that
is an ideal
(1) 0 Ii , i
[Since each Ii is an ideal of X, i
. By definition(1.18)]
0
(2) let x*y
and
y
x Ii , i
x*y Ii and y
[Since each Ii is ideal,
Ii , i
i
. By definition(1.18)]
x
Therefore ,
is an ideal of X.
To prove that
Let x
is a b-closed ideal
,then x Ii , i
8
b*(0*x)
Ii , i
[Since Ii is b-closed ideal, i
. By definition(2.20)]
b*(0*x)
is a b-closed ideal of X.
Theorem (2.26):
Let { Ii , i
b-closed ideal of X,
} be a chain of b-closed ideals of a BH-algebra X. then
is a
Proof
To prove that
is an ideal
(1) 0 Ii , i
[Since each Ii is an ideal of X, i
. By definition(1.18)]
0
(2) let x*y
and
y
Ij , Ik { Ii }i , such that x*y Ij and y Ik ,
either Ij Ik
or
Ik Ii
[ Since {Ii}i is a chain ]
either x*y Ij and y Ij
or
x*y Ik and y Ik
either x Ij
or
x Ik [ Since Ij and Ik are ideals. By definition(1.18)]
x
is an ideal
To prove that
let x
is a b-closed ideal of X.
Ij
b*(0*x)
{ Ii } i
Ij
such that x
Ij
[Since Ij is a b-closed ideal of X. By definition(2.20)]
b*(0*x)
is a b-closed ideal of X.
9
conclusion
In this research we generalize some basic concept from a BCH-algebra to a BH-algebra
firstly. Then we are bind between the BH-algebras and the abstract algebras. Finally we
define the concept of a closed ideal with respect to an element of a BH-algebra and give some
theorems about this notion.
References
[1] A. B. Saeid, A. Namdar and R.A. Borzooei, "Ideal Theory of BCH-Algebras",
World Applied Sciences Journal 7 (11): 1446-1455, 2009.
[2] C. H. Park, "Interval-valued fuzzy ideal in BH-algebras", Advance in fuzzy set and
systems 1(3), 231 240, 2006.
[3] H. H. Abbass and H. M. A. Saeed, "on closed BCH-algebra with respect to an
element of a BCH-algebra", Kufa Journal for Maths. and Computer Science, no 4,
2011.
[4] M. A. Chaudhry, "On BCH-algebra", Math. Japonica 36, 665-676, 1991.
[5] M.A. Chaudhry and H. Fakhar-Ud-Din, " Ideals and filters in BCH-algebra", Math.
japonica 44, No. 1, 101-112, 1996.
[6] Q. G. Yu, Y. B. Jun and E. H. Roh, "Special subsets in BH-algebras", Scientiae
Mathematica 2(3) , 311 314, 1999.
[7] Q. P. Hu and X. Li, "On BCH-algebras", Math. Seminar Notes Kobe University No.
2, Part 2, 11: 313-320, 1983.
[8] Q. Zhang, Y. B. Jun and E. H. Roh, "On the branch of BH-algebras", Scientiae
Mathematica Japonica 54(2), 363 367, 2001.
[9] Q. Zhang, E. H. Roh and Y. B. Jun, "On fuzzy BH-algebras", J. Huanggang, Normal
Univ. 21(3), 14 19, 2001.
[10] Y. B. Jun, E. H. Roh and H. S. Kim, "On BH-algebras", Scientiae Mathematica
1(1), 347 354, 1998.
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