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
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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