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 Total Downloads : 414
 Authors : Sisir Kumar Rajbongshi, Dr. Dwiraj Talukdar
 Paper ID : IJERTV2IS100261
 Volume & Issue : Volume 02, Issue 10 (October 2013)
 Published (First Online): 21102013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Some Properties of Fuzzy Boolean algebra
Some Properties of Fuzzy Boolean algebra Sisir Kumar Rajbongshi 1, Dr. Dwiraj Talukdar 2 Department of Information Technology, Gauhati University Guwahati781014, Assam, India
Abstract
The aim of this article is to continue the study of the behaviors of the fuzzy Boolean algebra formed by the fuzzy subsets of a finite set that has been introduced in [11]. In this article, the atom and the coatom of those fuzzy Boolean algebras are introduced and their properties are discussed.
Keywords
Fuzzy Boolean algebra, subelement, atom, coatom

Introduction
In [11], a kind of family of fuzzy subsets of a finite set had been introduced which can form fuzzy Boolean algebra, where the complement operation on the fuzzy subset was redefined. Further, those works had been extended by observing the properties of homomorphism, isomorphism and automorphism of the fuzzy Boolean algebra in [12]. The characteristics of ideals and filters had also been observed in that article.
The set of atoms forms a basis of the Boolean algebra because all elements can be expressed using these elements.
The aim of this article is to continue the study of
Every Boolean algebra has a trivial subelement or subset namely the set0 , consisting the bottom
element 0 alone; all other subelements or subsets of B are called nontrivial. Every Boolean algebra B has an improper subset namely B itself; all other subsets are called proper.
An atom of a Boolean algebra is an element which does not have nontrivial proper subelements. This implies that an element q of a
Boolean algebra B is an atom if q 0 and if there are only two elements p such that p q , namely 0 and q . Atoms are the elements which covers 0 . Atoms are also the immediate successor of 0.
A coatom of a Boolean algebra is an element
which does not have any proper superset. This implies that an element c of a Boolean algebra B is a coatom if c 0 and if there are only two elements d such that c d , namely 1 and c . Co atoms of a Boolean algebra are the elements which are the immediate predecessor of 1.
The main concept of fuzzy Boolean algebra which has been introduced in the article [11] is as follows:
For a finite set E x0 , x1 , x2 …, xn1 with n
elements with the set M of membership values, such that,
the behaviors of the fuzzy Boolean algebra formed
by the fuzzy subsets of a finite set which was
M 0, 1 , 2 , 3 ,…,
p p p
p 1 p
,
,
p p 1
introduced in [11]. In the first section, the definition of atom for a fuzzy Boolean algebra is introduced and then some properties of those atoms
0, h, 2h, 3h,…, ( p 1)h, ph, 1
are discussed. The next section is concerned with
the coatoms for a fuzzy Boolean algebra and their
where, h
p
and p is any number
properties are discussed.

Preliminaries
This section lists some basic definitions and concepts of Boolean algebra as follows:
Considering B be an arbitrary Boolean algebra and let p0 be an arbitrary element of B . Then the
set of elements p with p p0 or equivalently, the
Then for any mapping E 0, kh, where 1 k p , forms a Boolean algebra. This is
called fuzzy Boolean algebra, as it is formed by fuzzy subsets. The fuzzy subset consisting of all membership value equal to ' kh ' is called the universal fuzzy subset and the fuzzy subset consisting all membership value equal to '0 ' is called the empty fuzzy subset.
set of all elements of the form set of subelements of p0 .
p p0 is called the
Hence, for the mappings E 0, h, E 0, 2h,…E 0, ph,
p numbers of Fuzzy Boolean algebras can be obtained which are denoted by B1 , B2 , B3 ,…Bp respectively. This set of fuzzy
B1 [0 { x0 , 0, x1 , 0, x2 , 0},
1 { x0 , 0, x1 , 0, x2 , h},
4 { x , 0, x , h, x , 0},
1 2 3
1 2 3
p
p
Boolean algebras is denoted as B , which is: B B , B , B ,….B .
The scalar multiplication among the fuzzy Boolean algebras is defined based on the membership values of the elements of the fuzzy subsets. For, any two fuzzy Boolean algebras Br , Bs B, where, 1 r p,1 s p and r s , the scalar multiplication is defined as:
x r x ,x E ,
Br i s Bs i i
r
r
Where, i 0,1,…n. again, B xi and
0 1 2
5 { x0 , 0, x1 , h, x2 , h},
16 { x0 , h, x1 , 0, x2 , 0},
17 { x0 , h, x1 , 0, x2 , h},
20 { x0 , h, x1 , h, x2 , 0},
21={ x0 , h, x1 , h, x2 , h}]
Where, '0 ' is the empty fuzzy subset or the bottom element and 21 is the universal fuzzy subset or the top element of the fuzzy Boolean algebra B1 .
The Hass diagram of the fuzzy Boolean algebra
s
s
B xi are the membership values of the
B1 is shown in the Fig.1 bellow:
ith element of the fuzzy subsets of
Bs respectively.
Br and
0

Definition of Atom for a Fuzzy Boolean algebra
Every fuzzy Boolean algebra F has a trivial
subelement or fuzzy subset namely the set , consisting the empty fuzzy subset alone; all other fuzzy subsets of F are called nontrivial. Any fuzzy Boolean algebra F has an improper fuzzy
1 4 16
5 17 20
subset namely ' g ' which is called the universal
fuzzy subset or the top element; all other fuzzy subsets are called proper.
An atom of a fuzzy Boolean algebra F is an element or fuzzy subset which does not have non trivial proper fuzzy subsets. Alternatively, an atom of a fuzzy Boolean algebra F is a fuzzy subset which cannot be expressed as the join (fuzzy union) of other nontrivial proper fuzzy subsets. This means that an element v of a fuzzy Boolean algebra F is an atom if v and if there are only
two elements u such that u v , namely and v .
21
Fig.1 The Hass diagram of B1
Here, the atoms are 1, 4 and 16 because only these fuzzy subsets cannot be expressed as the join of other nontrivial proper fuzzy subsets.
In the following, some characterizations of the atoms of the fuzzy Boolean algebras have been discussed:
Hence, the atoms of a fuzzy Boolean algebra are1.1 3.2 Lemma
the elements which cover the empty fuzzy subset. Atoms are also the immediate successor of the empty fuzzy subset, .
The number of atoms of a fuzzy Boolean algebra is always equal to the number of elements in the universal set.
Proof: For a finite set E x0 , x1 , x2 …, xn1
3.1 Example
Let, E x0 , x1 , x2 be the universal set. Let,
1
with n elements and a set M of membership values, such that,
M 0, 1 , 2 , 3 ,…, p 1, p
M 0, h, 2h,3h 1 is a set, where h .
p p p p p 1
3
Now, considering a mapping from E to 0, h,
0, h, 2h, 3h,…, ( p 1)h, ph,
we get a fuzzy Boolean algebra B1 , written as follows:
where, h 1
p
and p is any positive integer.
Then for any mapping E 0, kh, where 1 k p , forms a fuzzy Boolean algebra.
So, by the type of mappings considered, we get exactly nfuzzy subsets containing exactly one non zero membership; which cannot be expressed as the join of nontrivial proper fuzzy subsets. Therefore, only those uzzy subsets are the atoms for a fuzzy Boolean algebra.
Hence, the number of atoms of a fuzzy Boolean algebra is always equal to the number of elements in the universal set.
Again, for the mappings E 0, h, E 0, 2h,…E 0, ph,
p numbers of Fuzzy Boolean algebras can be
obtained. Therefore, the total number of atoms that can be obtained from all the fuzzy Boolean algebras formed by the fuzzy subsets of a finite set is always equal to n p.

Lemma
In any fuzzy Boolean algebra, the atoms are the fuzzy subsets which contain exactly one nonzero membership element.
Proof: From the definition of fuzzy set union operation and from the fuzzy Boolean algebra that have been introduced in [11], it can be observed that only those fuzzy subsets which contain exactly one nonzero membership element cannot be
A [ A0 x0 , 0, x1 , 0,…, xn2 , 0, xn1 , kh, A1 x0 , 0, x1 , 0,…, xn2 , kh, xn1 , 0,
……………………………………………………..
……………………………………………………..,
An2 x0 , 0, x1 , kh,…, xn2 , 0, xn1 , 0,
An1 x0 , kh, x1 , 0,…, xn2 , 0, xn1 , 0].
Now, the supremum of all the atoms is the join (fuzzy union) of all the atoms, which is as follows:
A0 A1 A2 …. An1 =
x0 , kh,x1 , kh,…,xn2 , kh, xn1, kh=universa l fuzzy subset.
Again, the infimum of all the atoms is the meet
(fuzzy intersection) of all the atoms, which is as follows:
A0 A1 A2 …. An1 =
x0 , 0,x1 , 0,…,xn2 , 0,xn1 , 0 =empty fuzzy subset.
Therefore, is clear the empty fuzzy subset of a
fuzzy Boolean algebra is the infimum of the set of all atoms; on the other hand the universal fuzzy subset is the supremum of the set of all atoms.
expressed as the join of non trivial proper fuzzy1.2 3.6 Lemma
subsets; all the other non trivial proper fuzzy subsets can be expressed as the join of other non trivial proper fuzzy subsets. Hence, these fuzzy subsets are the only atoms of a fuzzy Boolean algebra.

Lemma
A fuzzy Boolean algebra is isomorphic to the power set of atoms by the mapping which maps each element of the fuzzy Boolean algebra to the set of atoms it dominates.
Proof: The proof is illustrated in the following example:
All the fuzzy Boolean algebras formed by the fuzzy1.3 3.7 Example
subsets of a finite set have the same number of atoms.
Proof: In the lemma 3.2, it is proved that the number of atoms of a fuzzy Boolean algebra is always equal to the number of elements in the universal set. Therefore, it is obvious that all the fuzzy Boolean algebra formed by the fuzzy subsets of the same universal set have the same number of atoms.

Lemma
The empty fuzzy subset of a fuzzy Boolean algebra is the infimum of the set of all atoms; on the other hand the universal fuzzy subset is the supremum of the set of all atoms.
Proof: Since, the atoms of a fuzzy Boolean algebra are the fuzzy subsets which contain exactly one nonzero membership, so, the set A of all atoms of a fuzzy Boolean algebra B is of the form as follows:
Considering the fuzzy Boolean algebra B1 as shown in example 3.1, where B1 0,1, 4,5,16,17, 20, 21 ,where A 1, 4,16 is
the set of atoms.
Now, defining a function f which takes each
element of B1 to the set of atoms it dominates, we get:
f 0 0, f 1 1, f 4 4, f 16 16,
f 5 1, 4, f 17 1,16, f 20 4,16,
f 21 1, 4,16
Again, the power set of X ,
A
0,1,4, 16, 1, 4, 1,16 , 4,16 , 1, 4,16
Hence, f is one to one and onto. So it is isomorphic.

Lemma
The set of atoms of a fuzzy Boolean algebra has onetoone correspondence to the set of atoms of another fuzzy Boolean algebra of the same universal set.
Proof: Considering a universal set
E x0 , x1 , x2 …, xn1 with n elements. From the


Definition of Coatom for Fuzzy Boolean algebra
Every fuzzy Boolean algebra F has an improper element or fuzzy subset namely the universal fuzzy subset ' g ' ; all other subsets are
called proper.
The coatoms of a fuzzy Boolean algebra are the
definition of atom of fuzzy Boolean algebra, the set fuzzy subsets which does not have any proper
of all atoms A of a fuzzy Boolean algebra
the following form:
A [ A x , 0, x , 0,…, x , 0, x
B1 is of
, kh,
fuzzy superset. Alternatively, a coatom of a fuzzy Boolean algebra is the fuzzy subset which cannot be expressed as the meet (intersection) of any
0 0 1
n2
n1
proper fuzzy subsets. This implies that an element
A1 x0 , 0, x1 , 0,…, xn2 , kh, xn1 , 0,
……………………………………………………..
……………………………………………………..,
An2 x0 , 0, x1 , kh,…, xn2 , 0, xn1 , 0,
An1 x0 , kh, x1 , 0,…, xn2 , 0, xn1 , 0].
or fuzzy subsets c of a fuzzy Boolean algebra F is said to a coatom if c g , where ' g ' is the
universal fuzzy subset and if there are only two elements or fuzzy subsets d and c such that c d , namely ' g ' and 'c ' .
Hence coatoms of a fuzzy Boolean algebra are the fuzzy subsets which are covered by the
Similarly, with the same universal set we can get another the set of atoms C of another fuzzy
universal fuzzy subset ' g '
immediate predecessor of ' g ' .
or which are the
Boolean algebra B2 of the form as follows:
C [C0 x0 , 0, x1 , 0,…, xn2 , 0, xn1 , k1h,
The coatoms of a fuzzy Boolean algebra is
illustrated in the following numerical example
C1 x0 , 0, x1 , 0,…, xn2 , k1h, xn1 , 01,.4 4.1 Example
……………………………………………………..
……………………………………………………..,
Cn2 x0 , 0, x1 , k1h,…, xn2 , 0, xn1 , 0,
Considering the fuzzy Boolean algebra B1 once again as in example 3.1, where B1 is written as follows:
B [0 { x , 0, x , 0, x , 0},
Cn1 x0 , k1h,x1, 0,…, xn2 , 0, xn1, 0].
Now, from the scalar multiplication that has been defined in [12], we can define a function f from A to C such that, f A0 C0 , f A1 C1 ,….. f An1 Cn1 .
Hence, f one to one and onto.
Therefore, it is clear that the set of atoms of a
1 0 1 2
1 { x0 , 0, x1 , 0, x2 , h},
4 { x0 , 0, x1 , h, x2 , 0},
5 { x0 , 0, x1 , h, x2 , h},
16 { x0 , h, x1 , 0, x2 , 0},
17 { x0 , h, x1 , 0, x2 , h},
20 { x0 , h, x1 , h, x2 , 0},
21={ x0 , h, x1 , h, x2 , h}]
fuzzy Boolean algebra has onetoone correspondence to the set of atoms of another fuzzy Boolean algebra of the same universal set.

Lemma
The number of elements of a fuzzy Boolean
The Hass diagram of bellow:
B1 is shown in the Fig. 2.
0
algebra= 2 atom
Proof: Since, the number of elements in a fuzzy Boolean algebra= 2 E 2no.of elements in the universal set
Also, in case of fuzzy Boolean algebra, it is
clear that:
Number of atoms=Number of elements in the universal set.
Hence, the number of elements of a fuzzy Boolean algebra= 2 atom
1 4 16
5 17 20
21
Fig.2 The Hass diagram of B1
Here, from the definition, the coatoms of B1 are
A [ A0 x0 , 0, x1 , 0,…, xn2 , 0, xn1 , kh,
5, 17 and 20.
A x, 0, x , 0,…, x
, kh, x
, 0,
Some characteristics of the coatoms of fuzzy
1 0 1
n2
n1
Boolean algebra are discussed as follows:

Lemma
……………………………………………………..
……………………………………………………..,
A x , 0, x , kh,…, x , 0, x , 0,
In a fuzzy Boolean algebra, the coatoms are the
n2 0 1
n2
n1
fuzzy subsets which contain exactly one zero membership.
Proof: From the definition of fuzzy set intersection operation and from the fuzzy Boolean algebra have been introduced it can be observed that all the proper fuzzy subsets except those which contains exactly one zero membership can be expressed as the meet of proper fuzzy subsets. This implies that only these elements are covered by the element universal fuzzy subset ' g ' . Hence, only
these elements or fuzzy subsets are the only co atoms of a fuzzy Boolean algebra.

Lemma
The number of coatoms of a fuzzy Boolean algebra is always equal to the numbers of elements in the universal set.
Proof: For a finite set E x0 , x1 , x2 …, xn1
with n elements with the set M of membership values, such that,
p p p p p 1
p p p p p 1
M 0, 1 , 2 , 3 ,…, p 1, p
0, h, 2h, 3h,…, ( p 1)h, ph,
An1 x0 , kh, x1 , 0,…, xn2 , 0, xn1 , 0].
Again, since, the coatoms of a fuzzy Boolean algebra are the fuzzy subsets which contain exactly one zero membership, so, the set of all coatoms C of the fuzzy Boolean algebra F is of the form as follows:
C [C0 x0 , 0, x1 , 0,…, xn2 , 0, xn1 , kh,
C1 x0 , 0, x1 , 0,…, xn2 , kh, xn1 , 0,
……………………………………………………..
……………………………………………………..,
Cn2 x0 , 0, x1 , kh,…, xn2 , 0, xn1 , 0, Cn1 x0 , kh, x1 , 0,…, xn2 , 0, xn1 , 0].
From the definition of complementation it follows
that: A0 C0 , A1 C1 ,…. An1 Cn1 . Therefore, coatoms and atoms of a fuzzy Boolean algebra are the complement of each other.
4.5 Lemma
where, h 1
p
and p is any positive integer.
The empty fuzzy subset of a fuzzy Boolean algebra is the infimum of the set of all coatoms; on
Since, for any mapping E 0, kh, where 1 k p , forms a fuzzy Boolean algebra.
So, there are exactly nfuzzy subsets containing exactly one zero membership; which cannot be expressed as the meet of proper fuzzy subsets. Therefore, only these fuzzy subsets are the co atoms of a fuzzy Boolean algebra. Hence, the number of coatom of a fuzzy Boolean algebra is always equal to the number of elements in the universal set.
Therefore, the number of atoms and coatoms of any fuzzy Boolean algebra is the same.
4.4 Lemma
the other hand the universal fuzzy subset is the supremum of the set of all coatoms.
Proof: Since, the coatoms of a fuzzy Boolean algebra are the fuzzy subsets which contain exactly one zero membership, so, the set of all coatoms C of a fuzzy Boolean algebra B is of the form as follows:
C [C0 x0 , 0, x1 , 0,…, xn2 , 0, xn1 , kh,
C1 x0 , 0, x1 , 0,…, xn2 , kh, xn1 , 0,
……………………………………………………..
……………………………………………………..,
Cn2 x0 , 0, x1 , kh,…, xn2 , 0, xn1 , 0, C x , kh, x , 0,…, x , 0, x , 0].
The coatoms and atoms of a fuzzy Boolean
algebra are the complement of each other.
n1 0 1
n2
n1
Proof: Since, the atoms of a fuzzy Boolean algebra are the fuzzy subsets which contain exactly one nonzero membership, so, the set of all atoms A of a fuzzy Boolean algebra F is of the form as
Now, the supremum of all the coatoms is the join (fuzzy union) of all the coatoms, which is as follows:
C0 C1 C2 …. Cn1 =
follows:
x , kh,x , kh,…,x
, kh, x
, kh=universa
0 1
l fuzzy subset.
n2
n1
Again, the infimum of all the atoms is the meet (fuzzy intersection) of all the atoms, which is as follows:
C0 C1 C2 …. Cn1 =
x0 , 0,x1 , 0,…,xn2 , 0,xn1 , 0=empty fuzzy subset.
Therefore, it is clear that empty fuzzy subset of
a fuzzy Boolean algebra is the infimum of the set of all coatoms; on the other hand the universal fuzzy subset is the supremum of the set of all co atoms.
Again it is proved that the empty fuzzy subset of a fuzzy Boolean algebra is also the infimum of the set of all atoms and the universal fuzzy subset is the supremum of the set of all atoms. So, this is a correspondence between the set of all atoms and the set of all coatoms of a fuzzy Boolean algebra.

Lemma:
The set of coatoms of a fuzzy Boolean algebra has onetoone correspondence with the set of co atoms of another fuzzy Boolean algebra of the same finite set.
Proof: The proof is obvious from the definition of scalar multiplication.

Lemma:
A fuzzy Boolean algebra is isomorphic to the power set of coatoms by the mapping which maps each element of the fuzzy Boolean algebra to the set of coatoms it precedes.
Proof: Considering the fuzzy Boolean algebra B as shown in example1, where B 0,1, 4,5,16,17, 20, 21 ,where A 5,17, 20 is
the set of coatoms.
Now, defining a function f which takes each
element of B to the set of coatoms it precedes, we get:
f 0 0, f 1 5,17, f 4 5, 20,
f 16 17, 20, f 5 5, f 17 17,
f 20 20, f 21 5,17, 20
Again, the power set of X ,
A
0,5,17, 20, 5,17, 5, 20 , 17, 20 , 5,17, 20
Hence, f is one to one and onto. So, f is isomorphic.


CONCLUSIONS
This article introduces the concept of atoms and coatoms of the fuzzy Boolean algebras formed by the fuzzy subsets of a finite set that have been introduced in [11]. Further, some properties of the atoms and coatoms are discussed which can be a foundation. But there is a lot of potential growth in this direction.

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