Pure and Applied Mathematics Journal
Volume 4, Issue 5, October 2015, Pages: 225-232

Fuzzy Derivations BCC-Ideals on BCC-Algebras

Samy M. Mostafa, Mostafa A. Hassan

Department of mathematics, Faculty of Education, Ain Shams University Roxy, Cairo, Egypt

Email address:

(S. M. Mostafa)
(M. A. Hassan)

To cite this article:

Samy M. Mostafa,Mostafa A. Hassan. Fuzzy Derivations BCC-Ideals on BCC-Algebras.Pure and Applied Mathematics Journal.Vol. 4, No. 5, 2015, pp. 225-232. doi: 10.11648/j.pamj.20150405.14


Abstract: In the theory of rings, the properties of derivations are important. In [15], Jun and Xin applied the notion of derivations in ring and near-ring theory to BCI-algebras, and they also introduced a new concept called a regular derivation in BCI-algebras. They investigated some properties of its .In this manuscript, the concept of fuzzy left (right) derivations BCC-ideals in BCC-algebras is introduced and then investigate their basic properties. In connection with the notion of homomorphism, the authors study how the image and the pre-image of fuzzy left (right) derivations BCC-ideals under homomorphism of BCC-algebras become fuzzy left (right) derivations BCC-ideals. Furthermore, the Cartesian product of fuzzy left (right) derivations BCC-ideals in Cartesian product of BCC-algebras is introduced and investigated some related properties.

Keywords: BCC-Ideals, Fuzzy Left (Right)-Derivations, the Cartesian Product of Fuzzy Derivations


1. Introduction

In 1966 Iami and Iseki [13,14] introduced the notion of BCK-algebras .Iseki [11,12] introduced the notion of a BCI-algebra which is a generalization of BCK-algebra. Since then numerous mathematical papers have been written investigating the algebraic properties of the BCK / BCI-algebras and their relationship with other structures including lattices and Boolean algebras. A BCC-algebra is an important class of logical algebras introduced by Y. Komori [16] and was extensively investigated by many researcher’s see [1,3,4,5,6,7,8]. The concept of fuzzy sets was introduced by Zadeh [21]. O. G. Xi [20] applied the concept of fuzzy set s to BCK-algebras. In the theory of rings, the properties of derivations are important. In [15], Jun and Xin applied the notion of derivations in ring and near-ring theory to BCI-algebras, and they also introduced a new concept called a regular derivation in BCI-algebras. They investigated some of its properties, defined a d-derivation ideal and gave conditions for an ideal to be d-derivation. Two years later, Hamza and Al-Shehri [9,10] studied derivation in BCK-algebras, a left derivation in BCI-algebras and investigated a regular left derivation of BCI-algebras. C. Prabpayak, U. Leerawat [18] applied the notion of a regular derivation to BCC algebras and investigated some related properties.

In this paper, the authors consider the the concept of fuzzy left (right) derivations BCC-ideals in BCC-algebras and investigate some properties of it. Moreover, the concepts of the image and the pre-image of fuzzy left (right) derivations BCC-ideals under homomorphism of BCC-algebras is given and studies some its properties. The Cartesian product of fuzzy left (right) derivations BCC-ideals in Cartesian product of BCC-algebras is introduced and investigated some related properties.

2. Preliminaries

In this section, we recall some basic definitions and results that are needed for our work.

Definition 2.1 [16] A BCC-algebra (,, 0) is a non-empty set  with a constant 0 and a binary operation  such that for all ,,  satisfying the following axioms:

(BCC-1)

(BCC-2)  0 =.

(BCC-3)

(BBC-4)

(BCC-5)

Definition 2.2 [5] Let (,, 0) be a BCC-algebra, we can define a binary relation  on as,  if and only if = 0, this makesas a partially ordered set.

Proposition 2.3 [8] Let (,, 0) be a BCC-algebra. Then the following hold ,, .

1. 

2. 

3. 

4. 

For elements x and y of a BCC-algebra  denote .

Lemma 2.4 [8] Let be a BCC-algebra. Then the following hold ,.

1. 

2. 

Example 2.5 [8] Let = be a set in which the operation is defined as follows:

0 1 2 3
0 0 0 0 0
1 1 0 1 0
2 2 2 0 0
3 3 3 1 0

Then is a BCC-algebra.

Definition 2.6 [8] Let (,, 0) be a BCC-algebra and S be a non-empty subset of , then S is called subalgebra of X if  .

Definition 2.7 [8] Let be a BCC-algebra and A be a non-empty subset of  , then A is called ideal of X if it satisfied the following conditions:

1.  

2.  

Definition 2.8 [8] Let be a BCC-algebra and  be a non-empty subset of  , then  is called BCC- ideal of  if it satisfied the following conditions:

1.  

2.  

Definition 2.9 [6] Let be a BCC-algebra, a fuzzy set m in X is called a fuzzy subalgebra

if

Definition 2.10 [6] Let (,, 0) be a BCC-algebra, a fuzzy set  in  is called a fuzzy BCC-ideal of  if it satisfied the following conditions:

(F)

(F)  min

Definition 2.11 [8] Let is a BCC-algebra,  ,we denote .

Definition 2.12 [18] Let be a BCC-algebra. A map d: is called a left- right derivation (briefly ()-derivation) of  if

.

Similarly, a map d: is called a right- left derivation (briefly-derivation) of  if

d (.

A map d: is called a derivation of  if d is both a ()-derivation and a -derivation of .

Example 2.13 [18] Let = be a BCC-algebra, in which the operation is defined as follows:

0 1 2 3
0 0 0 0 0
1 1 0 1 0
2 2 2 0 0
3 3 3 1 0

Define a map d: by

=.Then it is clear that d is a derivation of .

Example 2.14 Let = be a BCC-algebra, in which the operation is defined as follows:

0 1 2 3
0 0 0 0 0
1 1 0 1 0
2 2 2 0 0
3 3 3 1 0

Define a map d: by

=

Then d is a -derivation of  but is not a ()-derivation of .

Definition 2.15 [18] Let be a BCC-algebra and d: be a map of a QS-algebra  , then d is called regular if d (0)=0.

Lemma 2.16 [18] A derivation d of BCC algebra  is regular.

Proposition 2.17 [18] Let  be a BCC-algebra with partial order  and let  be a derivation of  Then the following hold for all

1.  

2.  

3.  

4.  

5.  

6.   is a sub-algebra of

Definition 2.18 Let  be a BCC-algebra and  be a derivation of

Denote

Proposition 2.19 [18]  be a BCC-algebra and d be a derivation of  Then  is a subalgebra of

3. Fuzzy Derivations BCC-Ideals on BCC-Algebras

In this section, we will discuss and investigate a new notion is called fuzzy left (right) derivations BCC-ideals on BCC-algebras and study several basic properties which are related to fuzzy left (right) derivations BCC-ideals.

Definition 3.1 Let be a BCC-algebra. and  be a self map. A non-empty subset  of a BCC-algebra  is called left derivations BCC-ideal of

If it satisfies the following conditions:

1.  

2.  

Definition 3.2 Let (,,0) be a BCC-algebra. and  be a self map. A non-empty subset  of a BCC-algebra  is called right derivations BCC-ideal of

If it satisfies the following conditions:

1.  

2.  

Definition 3.3 Let be a BCC-algebra. and  be a self map. A non-empty subset  of a BCC-algebra  is called derivations BCC-ideal of

If it satisfies the following conditions:

1.  

2.  

Definition 3.4 Let be a BCC-algebra. and  be a self map. A fuzzy set  in is called a fuzzy left derivations BCC-ideal (briefly-derivation) of  if it satisfies the following conditions:

(F)

(FL)

Definition 3.5 Let be a BCC-algebra. and  be a self map. A fuzzy set  in is called a fuzzy right derivations BCC-ideal (briefly -derivation) of  if it satisfies the following conditions:

(F)

(FR)

Definition 3.6 Let be a BCC-algebra. and  be a self map.

A fuzzy set  in is called a fuzzy derivations BCC-ideal of  if it satisfies the following conditions:

(F)

(F)

Remark 3.7

1.   If  is fixed, the definitions (3.1., 3.2., 3.3.) gives the definition BCC-ideal.

2.   If  is fixed, the definitions (3.4., 3.5., 3.6.) gives the definition fuzzy BCC-ideal.

Example 3.8 Let = be a BCC-algebra, in which the operation is defined as follows:

0 1 2 3
0 0 0 0 0
1 1 0 1 0
2 2 2 0 0
3 3 3 1 0

Define a map: by

=

Define a fuzzy set  by    where    with . Routine calculations give that  is not fuzzy left (right)-derivations BCC-ideal of BCC-algebra.

Example 3.9 Let = be a BCC-algebra, in which the operation is defined as follows:

0 1 2 3 4 5
0 0 0 0 0 0 0
1 1 0 0 0 0 1
2 2 2 0 0 1 1
3 3 2 1 0 1 1
4 4 4 4 4 0 1
5 5 5 5 5 5 0

Define a map  by

=

Define a fuzzy set  by    

 where    with . Routine calculations give that  is fuzzy left (right)-derivations BCC-ideal of BCC-algebra.

Theorem 3.10 Let  be a fuzzy left derivations BCC-ideal of BCC-algebra .

1.   If

2.   If

3.   If  

4.   If

 1. Let  and since

hence  then

2. Let  then by Theorem 3.10.1, we get

3. Let  

then by theorem 3.10.1, we get  

4. Let  then

 

Proposition 3.11 The intersection of any set of fuzzy left derivations BCC-ideals of BCC-algebra  is also fuzzy left derivations BCC-ideal.

 Let  be a family of fuzzy left derivations BCC-ideals of BCC-algebra

then

 and

=Lemma 3.12 The intersection of any set of fuzzy right derivations BCC-ideals of BCC-algebra  is also fuzzy right derivations BCC-ideal.

 Clear

Theorem 3.13 Let μ be a fuzzy set in  then μ is a fuzzy left derivations BCC-ideal of if and only if it satisfies : α[0,1]), U(μ , α) ≠ φ implies U(μ ,α) is BCC-ideal of…(A), where U (μ , α) = {x X / μ (d(x)) ≥ α}.

 Assume that  is a fuzzy left derivations BCC-ideal of  let  be such that U and  such that U then  and so by (FL),

 hence U

Let U and

it follows from (FL) that

 

so that UHence U is

BCC-ideal of

Conversely, suppose that  satisfies (A), let  be such that

, taking  we have  and

it follows that Uand U this is a contradiction and therefore  is a fuzzy left derivations BCC-ideal of

Theorem 3.14 Let μ be a fuzzy set in  then μ is a fuzzy right derivations BCC-ideal of if and only if it satisfies : α[0,1]), U(μ , α) ≠ φ implies U(μ ,α) is BCC-ideal of…(A), where U (μ , α) = {x X / μ (d(x)) ≥ α}.

 Clear

Definition 3.15 Let  be a fuzzy derivations BCC-ideal of BCC-algebra  the BCC-ideals   are called level BCC-ideal of  

4. Image (Pre-image) of Fuzzy Derivations BCC-Ideals Under Homomorphism

In this section, we introduce the concepts of the image and the pre-image of fuzzy left and right derivations BCC-ideals in BCC-algebras under homomorphism of BCC-algebras

Definition 4.1 Let be a mapping from the set to a set  If  is a fuzzy subset of  then the fuzzy subset β of is defined by

Is said to be the image of m under

Similarly if  is a fuzzy subset of  then the fuzzy subset  in  (i.e. the fuzzy subset is defined by  is called the preimage of  under

Theorem 4.2 An onto homomorphic preimage of a fuzzy right derivations BCC-ideal is also a fuzzy right derivations BCC-ideal under homomorphism of BCC-algebras

 Let  be an onto homomorphism of BCC-algebras,  a fuzzy right derivations BCC-ideal of  and  the preimage of  under  then  Let  we have

Now let  then

.

The proof is completed.

Theorem 4.3 An onto homomorphic preimage of a fuzzy left derivations BCC-ideal is also a fuzzy left derivations BCC-ideal.

 Clear

Definition 4.4 [2] A fuzzy subset  of  has sup property if for any subset  of

there exist  such that,  

Theorem 4.5 Let  be a homomorphism between BCC-algebras  and  For every fuzzy left derivations BCC-ideal  in   is a fuzzy left derivations BCC-ideal of

 By definition

 and .We have to prove that

Let  be an onto a homomorphism of BCC-algebras, μ a fuzzy left derivations BCC-ideal of  with sup property and β the image of μ under f, since μ is a fuzzy left derivations BCC-ideal of  we have μ(d(0)) ≥ μ(d(x))

 x Note that 0 , where 0,  are the zero of and respectively. Thus,

  which implies that   ,, let

be such that

and

 

Then

=

=

Hence β is a fuzzy left derivations BCC-ideal of

Theorem 4.6 Let  be a homomorphism between BCC-algebras  and  For every fuzzy right derivations BCC-ideal  in   is a fuzzy right derivations BCC-ideal of  

 Clear

5. Cartesian Product of Fuzzy Left Derivations BCC-ideals

Definition 5.1 [2] A fuzzy  is called a fuzzy relation on any set  if  is a fuzzy subset

Definition 5.2 [2] If  is a fuzzy relation on a set  and is a fuzzy subset of  then  is a fuzzy relation on  if

Definition 5.3 [2] Let  and  be a fuzzy subset of a set  the Cartesian product of  and  is defined by

Lemma 5.4 [2] Let μ and β be a fuzzy subset of a set  then

(i) is a fuzzy relation on  

 

Definition 5.5 If  is a fuzzy derivations relation on a set  and  is a fuzzy derivations subset of  then  is a fuzzy derivations relation on  if

Definition 5.6 Let  and  be a fuzzy derivations subset of a set  the Cartesian product of  and  is defined by

Definition 5.7 If  is a fuzzy derivations subset of a set  the strongest fuzzy relation on  that is a fuzzy derivations relation on  is  given by

Lemma 5.8 [2] For a given fuzzy derivations subset of a set  let  be the strongest fuzzy derivations relation on  then for  we have

Proposition 5.9 For a given fuzzy derivations subset  of BCC-algebra  let be the strongest fuzzy derivations relation on  If  is a fuzzy derivations BCC-ideal of  then

 Since is a fuzzy derivations BCC-ideal of  it follows from (F) that

where  then

Remark 5.10 Let  and  be BCC-algebras,

we define  on  by

 then clearly  is a BCC-algebra.

Theorem 5.11 Let  and  be a fuzzy derivations BCC-ideals of BCC-algebra  then  is a fuzzy derivations BCC-ideal of  

 1.

2. Let  then

Hence  is a fuzzy derivations BCC-ideal of  

Analogous to theorem 2.2[17], we have a similar result for fuzzy derivations BCC-ideal, which can be proved in similar manner, we state the result without proof.

Theorem 5.12 Let  and  be a fuzzy derivations subset of BCC-algebra  Such that  is a fuzzy derivations BCC-ideal of  then

(i)  either  or 

(ii) if  then either or  

(iii)         if  then either or  

(iv)        either  or  is a fuzzy derivations BCC-ideal of

Theorem 5.13 Let  be a fuzzy derivations subset of  BCC-algebra  and let  be the strongest fuzzy derivations relation on  then  is a fuzzy derivations BCC-ideal of  if and only if  is a fuzzy derivations BCC-ideal of  

 Let  be a fuzzy derivations BCC-ideal of 1. From (F), we get

 

2.  we have from (F)

Hence  is a fuzzy derivations BCC-ideal of  

Conversely, let  be a fuzzy derivations BCC-ideal of

1.  we have

.

It follows that which prove (F).

2. Let then

In particular, if we take  then

. This prove (F)

Hence  be a fuzzy derivations BCC-ideal of

6. Conclusion

Derivation is a very interesting and important area of research in the theory of algebraic structures in mathematics. In the present paper, the notion of fuzzy left and right derivations BCC-ideal in BCC-algebra are introduced and investigated the useful properties of fuzzy left and right derivations BCC-ideals in BCC-algebras.

In our opinion, these definitions and main results can be similarly extended to some other algebraic systems such as BCI-algebra, BCH-algebra, Hilbert algebra, BF-algebra, J-algebra, WS-algebra, CI-algebra, SU-algebra, BCL-algebra, BP-algebra, Coxeter algebra, BO-algebra, PU- algebras and so forth.

The main purpose of our future work is to investigate:

(1) The interval value, bipolar and intuitionist fuzzy left and right derivations BCC-ideal in BCC-algebra.

(2) To consider the cubic structure left and right derivations BCC- ideal in BCC-algebra.

We hope the fuzzy left and right derivations BCC-ideals in BCC-algebras, have applications in different branches of theoretical physics and computer science.

Algorithm for BCC-algebras

Input (set,  binary operation)

Output ("is a BCC-algebra or not")

Begin

If  then go to (1.);

End If

If  then go to (1.);

End If

Stop: =false;

;

While  and not (Stop) do

If  then

Stop: = true;

End If

While  and not (Stop) do

If  then

Stop: = true;

End If

End While

End While

If Stop then

Output ("is not a BCC-algebra")

Else

Output ("is a BCC-algebra")

End If

End.


References

  1. S. M. Bawazeer, N. O. Alshehri, and Rawia Saleh Babusail, "Generalized Derivations of BCC-Algebras,"International Journal of Mathematics and Mathematical Sciences, volume 2013, Article ID 451212, 4 pages.
  2. P. Bhattacharye and N. P. Mukheriee, Fuzzy relations and fuzzy group inform, sci, 36(1985), 267-282.
  3. W. A. Dudek, Y. B. Jun, Zoran Stojakovic, "On fuzzy ideals in BCC-algebras," Fuzzy Sets and Systems 123 (2001) 251-258.
  4. W. A. Dudek.The number of subalgebras of finite BCC-algebras,Bull. Inst. Math. Acad. Sinica, 20 (1992), 129–136.
  5. W. A. Dudek., On proper BCC-algebras, Bull. Inst. Math. Acad. Sinica, 20 (1992), 137–150.
  6. W. A. Dudek. and Y. B Jun, , Fuzzy BCC-ideals in BCC-algebras, Math. Montisnigri, 10 (1999), 21–30.
  7. W. A. Dudek. and Y. B Jun and C. Z .Stojakovi_, On fuzzy ideals in BCCalgebras,Fuzzy Sets and Systems, 123 (2001), 251–258.
  8. W. A. Dudek, and X.H Zhang, On ideals and congruences in BCCalgebras, Czechoslovak Math. J., 48 (123) (1998), 21–29.
  9. A. S. A Hamza and N. O. Al-Shehri. 2006. Some results on derivations of BCI-algebras. Coden Jnsmac 46: 13-19.
  10. A. S. A Hamza and N. O. Al-Shehri. 2007. On left derivations of BCI-algebras. Soochow Journal of Mathematics 33(3): 435-444.
  11. Y. Huang, BCI-algebra, Science Press, Beijing, 2006.
  12. K. Is´eki, "On BCI-algebras," Mathematics Seminar Notes, vol. 8, no. 1, pp. 125–130, 1980.
  13. K. Is´eki and S. Tanaka, "An introduction to the theory of BCKalgebras," Mathematica Japonica, vol. 23, no. 1, pp. 1–26, 1978.
  14. K. Is´eki and S. Tanaka, "Ideal theory of BCK-algebras," Mathematica Japonica, vol. 21, no. 4, pp. 351–366, 1976.
  15. Y. B. Jun, X. L. Xin. 2004.On derivations ofBCI-algebras. Information Sciences 159:167-176.
  16. Y. Komori, The class of BCC-algebras is not a variety, Math. Japonica, 29 (1984), 391–394.
  17. D. S. Malik and J. N. Mordeson, Fuzzy relation on rings and groups, Fuzzy Sets and Systems 43 (1991) 117-123.
  18. C. Prabpayak, Um Leerawat,On Derivations of BCC-algebras.Kasetsart J. (Nat. Sci.) 43: 398 - 401 (2009).
  19. A. Wro_nski, BCK-algebras do not form a variety, Math.Japonica, 28 (1983), 211–213.
  20. O. G. Xi, Fuzzy BCK-algebras, Math. Japon.36 (1991), 935 942.
  21. L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338–353.

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