Some Properties of Fuzzy Boolean algebra

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 Guwahati-781014, 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 co-atom of those fuzzy Boolean algebras are introduced and their properties are discussed.

Keywords

Fuzzy Boolean algebra, subelement, atom, co-atom

1. 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 non-trivial. 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 non-trivial 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 co-atom 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 co-atom 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 co-atoms for a fuzzy Boolean algebra and their

   where, h

   p

   and p is any number

   properties are discussed.

2. 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

3. 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 non-trivial. 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 non-trivial 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 non-trivial 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 n-fuzzy subsets containing exactly one non- zero membership; which cannot be expressed as the join of non-trivial 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.

   1. Lemma

      In any fuzzy Boolean algebra, the atoms are the fuzzy subsets which contain exactly one non-zero 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 non-zero 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.

   2. 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.

   3. 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 non-zero 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.

   1. Lemma

      The set of atoms of a fuzzy Boolean algebra has one-to-one 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

4. Definition of Co-atom 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 co-atoms 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 co-atom 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 co-atom 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 co-atoms 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 co-atoms 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 one-to-one correspondence to the set of atoms of another fuzzy Boolean algebra of the same universal set.

   1. 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 co-atoms 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 co-atoms of fuzzy

   1 0 1

   n2

   n1

   Boolean algebra are discussed as follows:

   1. Lemma

      ……………………………………………………..

      ……………………………………………………..,

      A x , 0, x , kh,…, x , 0, x , 0,

      In a fuzzy Boolean algebra, the co-atoms 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.

   2. Lemma

   The number of co-atoms 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 co-atoms of a fuzzy Boolean algebra are the fuzzy subsets which contain exactly one zero membership, so, the set of all co-atoms 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, co-atoms 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 co-atoms; on

   Since, for any mapping E 0, kh, where 1 k p , forms a fuzzy Boolean algebra.

   So, there are exactly n-fuzzy 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 co-atom of a fuzzy Boolean algebra is always equal to the number of elements in the universal set.

   Therefore, the number of atoms and co-atoms 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 co-atoms.

   Proof: Since, the co-atoms of a fuzzy Boolean algebra are the fuzzy subsets which contain exactly one zero membership, so, the set of all co-atoms 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 co-atoms 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 non-zero 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 co-atoms is the join (fuzzy union) of all the co-atoms, 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 co-atoms; 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 co-atoms of a fuzzy Boolean algebra.

   1. Lemma:

      The set of co-atoms of a fuzzy Boolean algebra has one-to-one 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.

   2. Lemma:

   A fuzzy Boolean algebra is isomorphic to the power set of co-atoms by the mapping which maps each element of the fuzzy Boolean algebra to the set of co-atoms 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 co-atoms.

   Now, defining a function f which takes each

   element of B to the set of co-atoms 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.

5. CONCLUSIONS

   This article introduces the concept of atoms and co-atoms 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 co-atoms are discussed which can be a foundation. But there is a lot of potential growth in this direction.

6. REFERENCES

1. Byliski, C. (1990). Functions from a set to a set. Formalized Mathematics, 1(1), 153-164.

2. Dubois, Didier J. Fuzzy sets and systems: theory and applications. Vol. 144. Access Online via Elsevier, 1980.

3. Frink, O. (1954). Ideals in partially ordered sets. The American Mathematical Monthly, 61(4), 223-234.

4. Givant, S., Halmos, P.: Introduction to Boolean Algebras. Undergrad. Texts Math. Springer, New York (2009).

5. Klir, George J., and Bo Yuan. Fuzzy sets and fuzzy logic. New Jersey: Prentice Hall, 1995.

6. Monk, J. D., & Bonnet, R. (1989). Handbook of Boolean algebras (Vol. 3). North Holland.

7. Rosen, K. H., & Krithivasan, K. (1999). Discrete mathematics and its applications (Vol. 6). New York: McGraw-Hill.

8. Sikorski, R., Sikorski, R., Sikorski, R., Sikorski, R., & Mathematician, P. (1969). Boolean algebras (Vol. 2). New York: Springer.

9. Squires, R. J. An introduction to Boolean algebra. Journal of the Institute of Actuaries Students Society 17 (1964):314.

10. Stone, M. H. (1936). The theory of representation for Boolean algebras. Transactions of the American Mathematical Society, 40(1), 37- 111.

11. Sisir Kumar Rajbongshi, and Dwiraj Talukdar, Some Aspects of fuzzy Boolean algebras formed by fuzzy subsets, IJARCSSE, 3.7 (2013): 1-8. Dwiraj Talukdar, and Sisir Kumar Rajbongshi. "An

12. Introductin to a Family of Fuzzy Subsets forming Boolean algebra." International Journal of Computer Applications 68.24 (2013): 1-6.