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ABSTRACT
The concepts of left (right) fuzzy derivations of d-ideals of d-algebra is introduced. The cartesian product of left (right) fuzzy derivations of d-ideals are investigated. Different characterizations of right (left) fuzzy derivation of ideals of d-algebra are discussed.
1. Introduction
After the introduction of fuzzy set theory by Zadeh (Citation1965), different fuzzification of the concepts of crisp set to fuzzy set become the major results. Jana et al. (Citation2017) introduced t-derivations on a complicated subtraction algebras and Mostefa, Abd-Elnaby, and Yousef (Citation2011) initiated the concept of fuzzy left (right) derivations of Ku-ideal of Ku-algebras. Senapati et al. (Citation2019, Citation2021) initiated the idea of cubic intuitionistic sub-algebras and closed cubic intuitionistic ideals of B-algebras and cubic intuitionistic structures applied to ideals of BCI-algebra. The notion of fuzzy left (right) derivations of BCC-ideals in BCC-algebras, and the Cartesian product of fuzzy left (right) derivations of BCC-ideals introduced by Jun et al. (Citation1999). Gerima and Fasil (Citation2020) introduced the concept of derivations in a BF-algebra. In addition, a left-right and a right-left derivation of BF2-algebra, left and right derivation of ideal in BF-algebras were discussed. The notion of d-algebras and d-ideals was introduced by Neggers and Kim (Citation1999). Left (right)-derivation of d-ideal of d-algebra was discussed by Young Hee kim (1018), and Akram and Dar (Citation2005) introduced the idea of fuzzy d-algebras. This concept was extended to structure of Fuzzy dot d-sub-algebras by Gerima Tefera (Citation2020). The concept of almost distributive fuzzy lattices with different characterization was introduced by Berhanu et al. (Citation2022) and characterization of homomorphism in implication algebra intiated by Tefera (Citation2022). The concepts of fuzzy derivations of ideals of BF-algebra with different properties discussed by Gerima and Tsige (Citation2022). These mentioned ideas motivated us to introduced the concepts fuzzy derivation in d-algebra as a new concept.
2. 2.Preliminaries
Definition 2.1.
Mostefa, Abd-Elnaby, and Yousef (Citation2011) An Algebra of type
is called a
algebra if it satisfies the following conditions:
and
implies
for all
Definition 2.2.
Neggers and Kim (Citation1999) A nonempty set with a constant 0 and a binary operation * is called a d-algebra, if it satisfies the following axioms:
and
implies
for all
Let be a non-empty subset of a d-algebra
, then S is called a sub-algebra of
if
for all
(Gerima Tefera Citation2020).
Definition 2.3.
(Neggers, Jun, and Kim Citation1999) Let be a
algebra and I be a subset of
, then I is called d-ideal of
if it satisfies following conditions:
and
implies
and
implies
, i.e.
Definition 2.4.
(Zadeh Citation1965) Let be a non-empty set. A fuzzy (sub)set
of the set
is a mapping
.
Definition 2.5.
(Akram and Dar Citation2005) A fuzzy set in d-algebra
is called a fuzzy sub-algebra of
if it satisfies
, for all
.
Definition 2.6.
(Hee Kim Citation2018) Let be a
algebra and let
for all
. The map
is said to be an
derivation if
for all
. Similarly, a map
said to be an
derivation if
for all
.
Definition 2.7
(Akram and Dar Citation2005) A fuzzy set in
is called fuzzy
ideal of
if it satisfies the following inequalities:
(Fd1)
(Fd2)
(Fd3) for all x, y ∈ X.
Definition 2.8.
(Jun and Xin Citation2004) Let be
algebra a fuzzy set
in
is called fuzzy derivation of
ideal of
if it satisfies the following conditions:
Definition 2.9.
(Akram and Dar Citation2005) Let be the fuzzy set of a set
. For a fixed
, the set
is called an upper level of
.
A fuzzy subset is called fuzzy relation on a set
, if
is a fuzzy subset
(Akram and Dar Citation2005) .
Definition 2.10.
(Akram and Dar Citation2005) If is a fuzzy relation on a set
and
is fuzzy subset of
, then
is a fuzzy relation on
if
.
3. Main Results
3.1. Fuzzy Derivatives of D-Ideals of D-Algebra
Definition 3.1.1.
Let be a
algebra and
be a self-map. A fuzzy subset
in
called a fuzzy right derivation of
ideal of X, if it satisfies the following conditions:
Example 3.1.1.
Let be a set and
be defined by the table below:
Table 3.1 Fuzzy right derivation of d-ideal of d-algebra
Then is
algebras.
Now define self-map by
And define a fuzzy derivation by
,
,
where
and
.
i.
Since
ii.
Then let ,
. So
.
Thus,
Let then
Thus,
Let then
But . Thus,
Let then
Note:
Let then
Let then
Let then
In any case, is fuzzy right derivation of
ideal of
algebra of
.
Remark 3.1.1.
In the above example (3.1.1) is not fuzzy left derivation of
ideal of
algebra of
.
Definition 3.1.2.
Let be a
algebra and
be a self--map. A fuzzy set
in
called a fuzzy left derivation of
ideal, if it satisfies the following conditions:
a.
b. for all
Example 3.1.2.
Let be a set and
be defined by the table below:
Table 3.2 Fuzzy left derivation of d-ideal of d-algebra.
Then is
algebras.
Now define self-map by
And define a fuzzy derivation by
,
where
and
.
Now by using the definition of fuzzy left derivations of ideal of
algebra
we can show that
is fuzzy left derivation of
ideal of
algebra
as follows:
i.
Since
ii.
Then let. So
Then let . So
Then let . So
Let . So
Let . So
But . Which is contradiction with
Thus, is not fuzzy left derivation of
ideal of
algebra of
.
Example 3.1.3.
Let be a set with * given by the following table:
Table 3.3 Left derivation of d – ideal of d – algebra
Then is
algebras.
Now define self-map by
And define a fuzzy derivation by
,
, where
and
.
i.
Since
In the same method it is easy to show the remaining part.
Hence is fuzzy left derivation of
ideal of
algebra of
.
Definition 3.1.3.
Let be fuzzy subset of
and
is
algebra. Let
, then
is level subset of
.
Definition 3.1.4. [4] Let be fuzzy set of a set
. For a fixed
, the set
is called an upper level of
.
Theorem 3.1.1.
Let be a fuzzy set in
then µ is a fuzzy left derivations of
ideal of
if and only if it satisfies : For all
implies
is
ideal of
where
.
Proof.
Let be a fuzzy subset in
(1). Assume that
be a fuzzy left derivations
ideal of
.
Let be a fuzzy left derivations of
ideal of
and
such that
and for
with
and
then
and
.
Hence,
(2).
Then
Since, is a fuzzy left derivation of
ideal of d-algebra
,
And hence, .
(3). Let and
we have to show
Now
Therefore,
And hence, is a
ideal of
.
Conversely, assume that satisfies
. Let
By taking
We have and
It follows that:
and
It contradicts and therefore is a fuzzy left derivations
ideal of
.
Theorem 3.1.2.
Let be a fuzzy set in
, then µ is a fuzzy right derivations of
ideal of
if and only if it satisfies : For all
implies
is
ideal of
where
.
Proof. It is similar with the prove of the above theorem (3.1.1). So, we omitted the proof.
Proposition 3.1.1.
The intersection of any family of fuzzy derivation of d-ideal of fuzzy left derivations ideal of
algebra
is also fuzzy left derivations
ideal.
Proof. Let be a family of fuzzy left derivations
ideals of
algebra
, then for any
.
Lemma 3.1.1.
The intersection of any family of fuzzy derivation of right derivations ideals of
algebra
is also fuzzy right derivations
ideal.
Definition 3.1.5.
Let and
be fuzzy left derivations subset of a set
, the Cartesian product of
and
is defined by
Definition 3.1.6 .
If is a fuzzy left derivations on a set
and
is a fuzzy left derivation on
if
.
Definition 3.1.7.
Let and
be fuzzy left derivations subset of a set
. Then the Cartesian product of
and
is defined by
Theorem 3.1.3.
Let and
be fuzzy left derivations
ideals of
algebra
, then
is a fuzzy left derivations
ideal of
Proof. for any we have
Now let then,
Hence, is a fuzzy left derivation
ideal of
Definition 3.1.8.
If is a fuzzy left derivations subset of a set
, the strongest fuzzy relation on
, that is a fuzzy derivation relation on
is
given by
.
Proposition 3.1.2.
For a given fuzzy subset of
algebra
, let
be the strongest left fuzzy derivation relation on
. If
is fuzzy left derivation
ideal of
, then
for all
.
Theorem 3.1.4.
Let be a fuzzy subset of
algebra
and let
be the strongest fuzzy left derivation
, then
is a fuzzy left derivation
algebra of
if and only if
left derivation
algebra of
.
Proof.
Assume that a fuzzy left derivation
algebra
, we note from
that
Now, for any we have from
:
Hence, is fuzzy left derivation
ideal of
.
Conversely, for all , we have
it follows:
which prove
.
Let then
In particular, if we take then,
This prove and complete the prove.
4. Conclusion
The notion of left (right) fuzzy derivations of d-ideals of d-algebra is introduced. The Cartesian product of left (right) fuzzy derivations of d-ideals is discussed. Strong fuzzy relations with illustrative examples are explained. Different characterizations theorems are proved. As a future work the authors indicate these idea can be extend to contra derivation of fuzzy ideals of d-algebra, Cubic intuitionistic sub algebras on d-algebra.
Disclosure statement
No potential conflict of interest was reported by the authors.
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