A Study of Generalized Fuzzy Dishkant Implications
DIMITRIOS S. GRAMMATIKOPOULOS, BASIL PAPADOPOULOS
Section of Mathematics and Informatics, Department of Civil Engineering
Democritus University of Thrace
Kimmeria 67100
GREECE
Abstract: - In this paper, we revisit the generalized Dishkant implications and provide analytical proof that they are
a new fuzzy implications’ class that contains the known class of Dishkant implications. Both classes are not always
fuzzy implications. For this reason we use the term operations instead of implications in general. Nonetheless,
it will be demonstrated that a necessary but not sufficient condition for a generalized Dishkant operation to be a
fuzzy implication exists. Furthermore, the intersection of the sets of generalized Dishkant operations and Dishkant
operations (respectively, implications) is provided. At the end, we prove a theorem for F- conjugation in GD-
operations.
Key-Words: - fuzzy negation, t- norm, t- conorm, D- implication
Received: May 15, 2023. Revised: August 18, 2023. Accepted: September 25, 2023. Published: October 9, 2023.
1 Introduction
As we said in [1], many fuzzy logic concepts are
derived from generalizations of classical tautologies.
Many classes of fuzzy implications and many of their
features are also such generalizations, as shown in [2],
[3], [4], [5], [6], [7], [8], [9], [10].
Fuzzy implications play an important part in many
applications, [11], [12], and are used in a wide range
of scientific areas. Fuzzy mathematical morphol-
ogy, approximate reasoning, image processing, con-
trol theories, expert systems, and others are examples.
In this research, we review and investigate GD-
implications, [1], a generalization from an existing
class of fuzzy implications known as Dishkant impli-
cations (abbreviated D- implications). The following
questions drove the inspiration for this study:
1. What happens if we are not restricted to use
only one fuzzy negation in a formula of a fuzzy
implication that contains a fuzzy negation, more
times than one time?
2. What are the results if we use different fuzzy
negations?
Indeed, using different fuzzy negations in such
formulas is not forbidden, [1], [6], [8], [9], [10], [13].
As a result, [1], introduces a new class of fuzzy impli-
cations known as generalized Dishkant implications
(shortly GD- implications). In this paper, we will
prove that this is a new class of fuzzy implications
as well as a hyper class of the known as D- implica-
tions’ class. Furthermore, this hyper class broadens
the required range of fuzzy implications.
The following is how the paper is structured: Sec-
tion 2 introduces the key principles for comprehend-
ing the article. Section 3 contains the analytical
proofs for the results we have presented in [1], some
examples that establish these results and the intersec-
tion of the sets of D- and GD- operations (respec-
tively, implications). We shall observe that GD- op-
erations are not necessarily fuzzy implications, and
we will provide a necessary but not sufficient con-
dition for a GD- operation to be a fuzzy implica-
tion. Furthermore, we will exclude some quadruples
(⊥,⊤,¬1,¬2)that do not produce GD- implications.
Finally, a theorem for F- conjugation in GD- oper-
ations will be demonstrated. Section 4 contains the
conclusions.
2 Preliminaries
Definition 1. [2], [14], [15], [16]. A decreasing
function ¬: [0,1] →[0,1] is called fuzzy negation, if
¬(0) = 1 and ¬(1) = 0. Moreover, a fuzzy negation
¬is called strong, if it is an involution, i.e.,
¬(¬(ε)) = ε, for all ε∈[0,1].
Remark 1. (i) The so called, least and greatest fuzzy
negations(see Example 1.4.4 in [2]) are respectively
¬0(ε) = 0,if ε > 0
1,if ε= 0 (1)
and
¬1(ε) = 0,if ε= 1
1,if ε < 1.(2)
(ii) We call ¬C(ε) = 1−εthe classical fuzzy negation,
which is a strong negation. Moreover, in this paper
we will use another type of fuzzy negations, which is
¬K(ε)=1−ε2(see Example 1.4.4 and Table 1.6 in
[2]).
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Definition 2. [2], [15], [16]. A function
⊤: [0,1] ×[0,1] →[0,1]
is called a t- norm, if it satisfies, ∀ε, ζ, δ ∈[0,1]:
⊤(ε, ζ) = ⊤(ζ, ε),(3)
⊤(ε, ⊤(ζ, δ)) = ⊤(⊤(ε, ζ), δ),(4)
ζ≤δ⇒ ⊤(ε, ζ)≤ ⊤(ε, δ),(5)
⊤(ε, 1) = ε. (6)
Dually, a t- conorm is a function
⊥: [0,1] ×[0,1] →[0,1]
if it satisfies, for all ∀ε, ζ, δ ∈[0,1], the above condi-
tions (3), (4), (5) and additionally
⊥(ε, 0) = ε. (7)
Remark 2. A t- norm we will use in this paper is
⊤P(ε, ζ) = ε·ζ(see Table 2.1 in [2]) and a t- conorm
is ⊥M(ε, ζ) = max{ε, ζ}(see Table 2.2 in [2]).
Definition 3. (See Definition 2.2.2 in [2]). We call a
t- conorm ⊥
(i) idempotent, if
⊥(ε, ε) = ε, ∀ε∈[0,1],(8)
(ii) positive, if
⊥(ε, ζ) = 1 ⇒ε= 1 or ζ= 1.(9)
Definition 4. [2], [16]. A t- conorm ⊥is strictly
monotone, if ⊥(ε, ζ)<⊥(ε, δ), whenever ε < 1and
ζ < δ.
Proposition 1. (See Proposition 9 in [4]). ∀ε, ζ ∈
[0,1]:
⊤(ε, ζ)≤ε≤ ⊥(ε, ζ)and ⊤(ε, ζ)≤ζ≤ ⊥(ε, ζ).
(10)
Remark 3. By Proposition 1, it follows that
⊥(1, ε) = ⊥(ε, 1) = 1, ε ∈[0,1] (11)
and
⊤(0, ε) = ⊤(ε, 0) = 0, ε ∈[0,1].(12)
Definition 5. (See Definition 2.3.8 in [2]). Let ¬be
a fuzzy negation and ⊥a t- conorm. We say that the
pair (⊥,¬)satisfies the law of excluded middle if
⊥(¬(ε), ε) = 1, ε ∈[0,1].(13)
Definition 6. [2], [17]. By Fwe denote the family of
all increasing bijections from [0,1] to [0,1]. We say
that functions λ, ν : [0,1]n→[0,1] are F- conjugate,
if there exists a f∈Fsuch that ν=λf, where for
any ε1, ε2,…, εn∈[0,1]:
λf(ε1, ε2,…, εn) = f−1(λ(f(ε1), f (ε2),…, f(εn))).
(14)
Remark 4. (See Proposition 1.4.8, Remarks
2.1.4(vii) and 2.2.5(vii) in [2]). It is easy to prove
that if f∈Fand ⊤is a t- norm, ⊥is a t- conorm
and ¬is a fuzzy negation (respectively strong), then
⊤fis a t- norm, ⊥fis a t- conorm and ¬fis a fuzzy
negation (respectively strong).
Definition 7. [2], [14]. A function
Σ : [0,1] ×[0,1] →[0,1]
is called a fuzzy implication if
Σ(·, ζ)is decreasing, (15)
Σ(ε, ·)is increasing, (16)
Σ(0,0) = 1,(17)
Σ(1,1) = 1,(18)
Σ(1,0) = 0.(19)
Remark 5. By axioms (16) and (17) we deduce the
normality condition
Σ(0,1) = 1.(20)
Moreover, by Definition 7 it is easy to prove the left
and right boundary conditions, [2]
Σ(0, ζ) = 1, ζ ∈[0,1],(21)
Σ(ε, 1) = 1, ε ∈[0,1].(22)
Definition 8. (See Definition 1.3.1 in [2]). A fuzzy
implication Σis said to satisfy the left neutrality prop-
erty, if
Σ(1, ζ) = ζ, ζ ∈[0,1],(23)
Remark 6. (i) Property (23) is not limited to fuzzy
implications, but in any function
Σ : [0,1] ×[0,1] →[0,1],
(ii) It is proved that, if f∈Fand
Σ : [0,1] ×[0,1] →[0,1],
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satisfies (15) (respectively (16), (17), (18), (19)), then
Σfis also satisfies (15) (respectively (16), (17), (18),
(19)). Moreover, if Σis a fuzzy implication, then Σf
is also a fuzzy implication (see Proposition 1.1.8 in
[2]).
Lemma 1. (See Lemma 1.4.14 in [2]). If a function
Σ : [0,1] ×[0,1] →[0,1],
satisfies (15), (17) and (19), then the function ¬Σ:
[0,1] →[0,1] is a fuzzy negation, where
¬Σ(ε) = Σ(ε, 0), ε ∈[0,1].(24)
Definition 9. (See Definition 1.4.15 in [2]). Let
Σ : [0,1] ×[0,1] →[0,1],
be a fuzzy implication. The function ¬Σdefined by
Lemma 1 is called the natural negation of Σ.
Definition 10. [2], [18], [19]. A function
Σ : [0,1] ×[0,1] →[0,1],
is called a D- operation if there exist a t-norm ⊤, a
t-conorm ⊥and a fuzzy negation ¬such that
Σ(ε, ζ) = ⊥(⊤(¬(ε),¬(ζ)), ζ), ε, ζ ∈[0,1].(25)
If Σis a D-operation generated from the triple
(⊤,⊥,¬), then we will often denote it by Σ⊤,⊥,¬.
Remark 7. [2], [18], [19]. D- operations are not
fuzzy implications in general since (16) could not
hold. Only if the D- operation is a fuzzy implication,
we will use the term D- implication.
3 Generalized Dishkant Implications
In this Section all the statements of [1], will be proved
in detail and supplemented with some more results
and a figure.
Definition 11. [1]. A function
Σ : [0,1] ×[0,1] →[0,1],
is called a GD- operation, if there exist a t- conorm
⊥, a t- norm ⊤and two fuzzy negations ¬1,¬2, such
that
Σ(ε, ζ) = ⊥(⊤(¬1(ε),¬2(ζ)), ζ), ε, ζ ∈[0,1].
(26)
If Σis a GD- operation generated by the quadruple
(⊥,⊤,¬1,¬2), then we denote it by Σ⊥,⊤,¬1,¬2.
Theorem 1. [1]. Σ⊥,⊤,¬1,¬2satisfies (15), (17), (18),
(19), (20) and (22). Furthermore ¬Σ⊥,⊤,¬1,¬2=¬1,
where ¬Σ⊥,⊤,¬1,¬2(ε) = Σ⊥,⊤,¬1,¬2(ε, 0), ε ∈[0,1].
Proof. Let Σ⊥,⊤,¬1,¬2be a GD- operation, then for
ε, ζ, δ ∈[0,1], if
ε≤ζ⇒ ¬1(ε)≥ ¬1(ζ)
(5)
⇒ ⊤(¬2(δ),¬1(ε)) ≥ ⊤(¬2(δ),¬1(ζ))
(3)
⇒ ⊤(¬1(ε),¬2(δ)) ≥ ⊤(¬1(ζ),¬2(δ))
(5)
⇒ ⊥(δ, ⊤(¬1(ε),¬2(δ))) ≥ ⊥(δ, ⊤(¬1(ζ),¬2(δ)))
(3)
⇒ ⊥(⊤(¬1(ε),¬2(δ)), δ)≥ ⊥(⊤(¬1(ζ),¬2(δ)), δ)
⇒Σ⊥,⊤,¬1,¬2(ε, δ)≥Σ⊥,⊤,¬1,¬2(ζ, δ),
which means that Σ⊥,⊤,¬1,¬2satisfies (15).
Σ⊥,⊤,¬1,¬2satisfies (17), since
Σ⊥,⊤,¬1,¬2(0,0) = ⊥(⊤(¬1(0),¬2(0)),0)
(7)
=⊤(¬1(0),¬2(0))
=⊤(1,1)
(6)
= 1.
Σ⊥,⊤,¬1,¬2satisfies (18), since
Σ⊥,⊤,¬1,¬2(1,1) = ⊥(⊤(¬1(1),¬2(1)),1)
=⊥(⊤(0,0),1)
(12)
=⊥(0,1)
(11)
= 1.
Σ⊥,⊤,¬1,¬2satisfies (19), since
Σ⊥,⊤,¬1,¬2(1,0) = ⊥(⊤(¬1(1),¬2(0)),0)
(7)
=⊤(¬1(1),¬2(0))
=⊤(0,1)
(6)
= 0.
Σ⊥,⊤,¬1,¬2satisfies (20), since
Σ⊥,⊤,¬1,¬2(0,1) = ⊥(⊤(¬1(0),¬2(1)),1)
(11)
= 1.
Σ⊥,⊤,¬1,¬2satisfies (22), since ∀ε∈[0,1]:
Σ⊥,⊤,¬1,¬2(ε, 1) = ⊥(⊤(¬1(ε),¬2(1)),1)
(11)
= 1.
Lastly, ∀ε∈[0,1] we have
¬Σ⊥,⊤,¬1,¬2(ε) = Σ⊥,⊤,¬1,¬2(ε, 0)
=⊥(⊤(¬1(ε),¬2(0)),0)
(7)
=⊤(¬1(ε),¬2(0))
=⊤(¬1(ε),1)
(6)
=¬1(ε).
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Proposition 2. [1]. Σ⊥,⊤,¬1,¬2satisfies (23).
Proof. ∀ζ∈[0,1] it is
Σ⊥,⊤,¬1,¬2(1, ζ) = ⊥(⊤(¬1(1),¬2(ζ)), ζ))
=⊥(⊤(0,¬2(ζ)), ζ))
(12)
=⊥(0, ζ)
(3)
=⊥(ζ, 0)
(7)
=ζ.
Thus, Σ⊥,⊤,¬1,¬2satisfies (23).
Remark 8. [1]. If ¬1=¬2the corresponding GD-
operation is a D- operation. Thus, GD- operations
sometimes do not satisfy (16). The same happens even
if we use different negations according to the follow-
ing Example 1. For these reasons, we use the term
GD- operations, instead of GD- implications.
Example 1. Consider the quadruple
(⊥M,⊤P,¬C,¬K). The corresponding GD-
operation is
Σ⊥M,⊤P,¬C,¬K(ε, ζ) = ⊥M(⊤P(¬C(ε),¬K(ζ)), ζ)
=⊥M(¬C(ε)· ¬K(ζ), ζ)
=⊥M((1 −ε)·(1 −ζ2), ζ)
=max{(1 −ε)·(1 −ζ2), ζ}
which is not a fuzzy implication, since
0.1≤0.2⇒Σ⊥M,⊤P,¬C,¬K(0.1,0.1) = 0.891 >
0.864 = Σ⊥M,⊤P,¬C,¬K(0.1,0.2).
Thus, Σ⊥M,⊤P,¬C,¬Kdoes not satisfy (16).
Proposition 3. [1]. If Σ⊥,⊤,¬1,¬2satisfies (16), then
we call it GD- implication.
Proof. The proof is obvious.
Proposition 4. [1]. Σ⊥,⊤,¬1,¬2satisfies (21) if and
only if the pair (⊥,¬2)satisfies (13).
Proof. If Σ⊥,⊤,¬1,¬2satisfies (21), then ∀ζ∈[0,1]:
Σ⊥,⊤,¬1,¬2(0, ζ) = 1 ⇒ ⊥(⊤(¬1(0),¬2(ζ)), ζ) = 1
⇒ ⊥(⊤(1,¬2(ζ)), ζ) = 1
(3)
⇒ ⊥(⊤(¬2(ζ),1), ζ) = 1
(6)
⇒ ⊥(¬2(ζ), ζ) = 1.
Thus, the pair (⊥,¬2)satisfies (13).
Conversely, if the pair (⊥,¬2)satisfies (13), then
∀ζ∈[0,1] it is
Σ⊥,⊤,¬1,¬2(0, ζ) = ⊥(⊤(¬1(0),¬2(ζ)), ζ)
=⊥(⊤(1,¬2(ζ)), ζ)
(3)
=⊥(⊤(¬2(ζ),1), ζ)
(6)
=⊥(¬2(ζ), ζ)
(13)
= 1.
Therefore, Σ⊥,⊤,¬1,¬2satisfies (21).
Corollary 1. [1]. If Σ⊥,⊤,¬1,¬2is a GD- implication,
then the pair (⊥,¬2)satisfies (13).
Proof. If Σ⊥,⊤,¬1,¬2is a GD- implication, then it sat-
isfies (21). So, by Proposition 4 we deduce that the
pair (⊥,¬2)satisfies (13).
Remark 9. [1]. (i) Corollary 1 gives a necessary, but
not sufficient condition for the generation of a GD-
operation Σ⊥,⊤,¬1,¬2. Note that every D- operation
(respectively implication) Σ⊤,⊥,¬is also a GD- oper-
ation (respectively implication) Σ⊥,⊤,¬,¬. Mas et al.
in Proposition 3 in [18] mention that if Σ⊤,⊥,¬is a D-
implication, then the pair (⊥,¬)satisfies (13), where
¬is a strong fuzzy negation. They also mention af-
ter Proposition 3 that this condition (they mean (13))
given in the previous proposition (i.e. Proposition 3
in [18]) is necessary but not sufficient. Moreover, we
must note that this proposition is proved for strong
fuzzy negations only, but the proof is similar and holds
for any fuzzy negation ¬.
(ii) By Corollary 1 it is obvious that, if the pair (⊥,¬2)
does not satisfy (13), i.e. ⊥(¬(ε), ε)= 1, for some
ε∈(0,1), then the obtained Σ⊥,⊤,¬1,¬2GD- opera-
tion is not a fuzzy implication.
Example 2. Consider the quadruple /..l; where ⊥and
⊤are any t- conorm and t-norm, respectively. The
corresponding GD- operation, which is a GD- impli-
cation (the proof is simple) is
I⊥,⊤,¬0,¬1(ε, ζ) = ⊥(⊤(¬0(ε),¬1(ζ)), ζ)
=⊥(⊤(¬0(ϵ),0),1),if ζ= 1
⊥(⊤(¬0(ϵ),1), ζ),if ζ < 1
(12)
=
(6) ⊥(0,1),if ζ= 1
⊥(¬0(ε), ζ),if ζ < 1
(11)
=1,if ζ= 1
⊥(0, ζ),if ε > 0and ζ < 1
⊥(1, ζ),if ε= 0 and ζ < 1
(3)
=
(11) 1,if ζ= 1
⊥(ζ, 0),if ε > 0and ζ < 1
1,if ε= 0 and ζ < 1
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(7)
=1,if ζ= 1
ζ, if ε > 0and ζ < 1
1,if ε= 0 and ζ < 1
=ζ, if ε > 0
1,otherwise
=1,if ε= 0
ζ, otherwise
=I12(ε, ζ).
See Figure 5 in page 509 in [3], for the formula of
I12.
Remark 10. By Remark 9 and Example 2 we deduce
that there are GD- implications, that are not D- im-
plications. Firstly, ¬Σ⊤,⊥,¬=¬Σ⊥,⊤,¬,¬=¬. More-
over, there does not exist any t-conorm ⊥such that
the pair (⊥,¬0)satisfies (13), since
⊥(¬0(0.3),0.3) = ⊥(0,0.3) (3)
=⊥(0.3,0) (7)
= 0.3= 1.
Thus, there does not exist any D- implication that
has ¬0as its natural negation. On the other hand
Σ⊥,⊤,¬0,¬1=I12 is a GD- implication with ¬0as its
natural negation, that means it is not a D- implication.
Therefore, the class of GD- implications is a new hy-
per class of that of D- implications, which contains
them.
These results lead us to the following Figure 1.
Figure 1: The intersection among the sets of D- opera-
tions (respectively, implications) and GD- operations
(respectively, implications).
Theorem 2. [1]. If ⊥is any idempotent, strict or pos-
itive t- conorm, ⊤is any t- norm, ¬1is any fuzzy nega-
tion and ¬2is any continuous fuzzy negation, then
Σ⊥,⊤,¬1,¬2is not a fuzzy implication.
Proof. Firstly, it has been proved (see Theorem 1.4.7
in [2]), there exists exactly one ξ∈(0,1), such that
¬(ξ) = ξ, where ¬is any continuous fuzzy negation.
If ⊥is strict t- conorm, then
⊥(¬(ξ), ξ) = ⊥(ξ, ξ)= 1,
because if
⊥(ξ, ξ) = 1 ⇔ ⊥(ξ, ξ) = ⊥(e, 1),
a contradiction, since
ξ < 1⇒ ⊥(ξ, ξ)<⊥(ξ, 1).
So, Σ⊥,⊤,¬1,¬2is not a fuzzy implication according
to Remark 9(ii). Furthermore, since the only idem-
potent, which is also a positive t- conorm is ⊥M(see
Remark 2.2.5(ii) and Table 2.2 in [2]), we will con-
tinue the proof only for positive t-conorms ⊥. If we
assume that ⊥is any positive t- conorm, then
⊥(¬(ξ), ξ) = S(ξ, ξ)= 1,
since ξ < 1. So, Σ⊥,⊤,¬1,¬2is not a fuzzy implication
according to Remark 9(ii).
Theorem 3. [1]. If f∈Fand Σ⊥,⊤,¬1,¬2
is a GD- operation (respectively implication), then
(Σ⊥,⊤,¬1,¬2)fis a GD- operation (respectively impli-
cation) and moreover
(Σ⊥,⊤,¬1,¬2)f= Σ⊥f,⊤f,(¬1)f,(¬2)f.
Proof. According to the Remark 6(ii) if Σ⊥,⊤,¬1,¬2
is a GD- operation (respectively implication), then
(Σ⊥,⊤,¬1,¬2)fis a GD- operation (respectively impli-
cation). Moreover, ∀ε, ζ ∈[0,1]:
(Σ⊥,⊤,¬1,¬2)f(ε, ζ) = f−1(Σ⊥,⊤,¬1,¬2(f(ε), f (ζ)))
=f−1(⊥(⊤(¬1(f(ε)),¬2(f(ζ))), f(ζ)))
=f−1(⊥(⊤(f(f−1(¬1(f(ε)))),
f(f−1(¬2(f(ζ))))), f(ζ)))
=f−1(⊥(⊤(f((¬1)f(ε)), f((¬2)f(ζ))), f(ζ)))
=f−1(⊥(f(f−1(⊤(f((¬1)f(ε)), f((¬2)f(ζ))))),
f(ζ)))
=⊥f(⊤f((¬1)f(ε),(¬2)f(ζ)), ζ)
= Σ⊥f,⊤f,(¬1)f,(¬2)f(ε, ζ).
4 Conclusion
There exist fuzzy implications, which have a fuzzy
negation function more than once in their formula. Is
the use of only one fuzzy negation in these formu-
lations binding? It is self-evident that the response
is negative. In this study, we revisit a hyper class
of the well-known class of fuzzy implications known
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as Dishkant implications. This hyperclass is known
as generalized Dishkant implications (or GD- impli-
cations for short). The downside of GD- implica-
tions is that they do not always satisfy (16). As a re-
sult, we refer to GD- operations rather than implica-
tions in general. The characterization of quadruples
(⊥,⊤,¬1,¬2), such that Σ⊥,⊤,¬1,¬2satisfies (16) re-
mains unsolved. The same problem holds for the
characterization of triples (⊤,⊥,¬), such that Σ⊤,⊥,¬
satisfies (16) (see page 108 in [2]).
On the other hand, it has been demonstrated that
the set of D- operations is a subset of the set of GD-
operations, and the findings are depicted in Figure 1.
It has been demonstrated that a necessary but not suf-
ficient condition for a GD- operation to be a fuzzy
implication exists (see Corollary 1 and Remark 9).
Theorem 2 excludes quadruples (⊥,⊤,¬1,¬2)that
do not generate GD- implications, and Theorem 3 in-
vestigates the relationship of F−conjugation in GD-
operations.
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Contribution of individual authors to
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Dimitrios S. Grammatikopoulos: Writing- original
draft.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.78
Dimitrios S. Grammatikopoulos, Basil Papadopoulos
E-ISSN: 2224-2880
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Volume 22, 2023
Basil Papadopoulos: Supervision.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.78
Dimitrios S. Grammatikopoulos, Basil Papadopoulos
E-ISSN: 2224-2880
718
Volume 22, 2023
