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Abstract
In this paper, we revisit and study the basic properties of two families of fuzzy implications, the so-called and
implications. More specific, we study when these fuzzy implications satisfy, or not, the neutrality property
, the exchange principle
, the identity principle
and the ordering property
. Moreover, a study is presented for the law of importation
with respect to a t- norm. Also, we study the relation of
conjugation in
implications.
1. Introduction
The generalization of the notion of implication from classical to fuzzy logic is a known process with its difficulties [Citation1,Citation2]. In this paper, we will study the properties of such generalizations, the so-called and
implications. These implications are the generalization of the following classical tautology
(1)
(1)
Firstly,
implications mentioned by many authors, such as Dubois and Prade [Citation3] in 1984, Fodor [Citation4,Citation5] in 1991, Fodor and Roubens [Citation2] in 1994 and Klement et al. [Citation6] in 2000. These authors addressed their correlation with another family, the so called
implications, when we use a strong negation N. Bedregal [Citation7] in 2007 defined them for any t-norm and any fuzzy negation. Baczyński and Jayaram [Citation1] in 2008 related them with R-implications, when the t-norm T is left continuous and N is a strong negation. Pradera et al. [Citation8] in 2016 mentioned the formula of
implications, using aggregation functions in general. Despite these references, they remained anonymous and unstudied until 2017.
In 2017, Pinheiro et al. [Citation9] in their homonymous paper named them implications. They studied them in 2017 [Citation9] and 2018 [Citation10] for strong and not, negations. Some more results on functional equations and
implications are presented by Pinheiro et al. in 2018 [Citation11]. In 2020 [Citation12], the intersection between
and
implications presented and moreover their close relation in a way that they simulate the same (or a similar) way of classical thinking. Additionally, since a fuzzy negation is not a uniquely determined function in 2018 Pinheiro at al. [Citation13] studied
implications, which are
implications generated from not necessary the same negation.
The aforementioned recent interest was the motivation of our study. In this paper, we revisit and
implications and we study whether or not they satisfy basic properties of fuzzy implications and the law of importation with respect to a t-norm [Citation1].
2. Preliminaries
Definition 2.1
[Citation1,Citation2,Citation6,Citation14]
A decreasing function is called fuzzy negation, if
and
. Moreover, a fuzzy negation N is called
strict, if it is continuous and strictly decreasing,
strong, if it is an involution, i.e.
and non-filling, if
Remark 2.1
The so-called crisp fuzzy negations ([Citation15, Remark 2.1]) are
(2)
(2)
(3)
(3)
Lemma 2.1
[Citation1, Lemma 1.4.9].
If , N are fuzzy negations such that
, then
(i) |
| ||||
(ii) | N is a strictly decreasing fuzzy negation. |
Definition 2.2
[Citation1,Citation6,Citation14]
A function is called a triangular norm (shortly t-norm), if it satisfies, for all
, the following conditions
(T1)
(T1)
(T2)
(T2)
(T3)
(T3)
(T4)
(T4)
Dually, a function
is called a triangular conorm (shortly t-conorm) if it satisfies, for all
, the above conditions (T1), (T2), (T3), which are denoted by (S1), (S2), (S3) and additionally
(S4)
(S4)
Definition 2.3
[Citation1,Citation6]
A t-norm T is strictly monotone, if , whenever x>0 and y<z.
Definition 2.4
[Citation1,Citation6]
A t-norm T is called continuous if it is continuous in both the arguments.
Definition 2.5
[Citation1, Definition 2.3.14].
Let T be a t-norm and N be a fuzzy negation. We say that the pair satisfies the law of contradiction if
(LC)
(LC)
Definition 2.6
[Citation1,Citation16]
By Φ we denote the family of all increasing bijections from to
. We say that functions
are Φ- conjugate, if there exists a
such that
, where
Remark 2.2
[Citation1, Propositions 1.4.8, Remarks 2.1.4(vii) and 2.2.5(vii)].
It is easy to prove that if and T is a t-norm, S is a t-conorm and N is a fuzzy negation (respectively strict, strong), then
is a t-norm,
is a t-conorm and
is a fuzzy negation (respectively strict, strong).
Definition 2.7
[Citation1,Citation2]
A function is called a fuzzy implication if
(I1)
(I1)
(I2)
(I2)
(I3)
(I3)
(I4)
(I4)
(I5)
(I5)
Definition 2.8
[Citation1,Citation10]
A fuzzy implication I is said to satisfy
the left neutrality property, if
(NP)
(NP)
the exchange principle, if
(EP)
(EP)
the identity principle, if
(IP)
(IP)
the ordering property, if
(OP)
(OP)
the left ordering property, if
(LOP)
(LOP)
the right ordering property, if
(ROP)
(ROP)
Remark 2.3
[Citation1, Proposition 1.1.8].
It is proved that, if and
is a fuzzy implication, then
is also a fuzzy implication.
Definition 2.9
[Citation1, page 223].
A fuzzy implication I is said to satisfy the law of importation with respect to a t-norm T, if
(LI)
(LI)
Remark 2.4
[Citation1, Remark 7.3.1].
It can be immediately seen that if a fuzzy implication I satisfies (LI) with respect to any t-norm T, by the commutativity of the t-norm T, we have that I satisfies (EP).
Definition 2.10
[Citation1, Lemma 1.4.14 and Definition 1.4.15].
Let be a fuzzy implication. The function
defined by
is called the natural negation of I.
Definition 2.11
[Citation1,Citation6]
A function is called an
implication if there exist a t-conorm S and a fuzzy negation N such that
(4)
(4)
Moreover, if I is an
implication generated from S and N, then we will often denote it by
.
Proposition 2.2
[Citation1, Definition 2.4.3(i)]
If is an
implication, then it satisfies (NP) and (EP).
3. ![](//:0)
and ![](//:0)
Implications
In this section we investigate the properties of and
implications.
Definition 3.1
[Citation10, Proposition 3.1 and Definition 3.1].
A function is called a
implication if there exist a t-norm T and a fuzzy negation N such that
(5)
(5)
Moreover, if I is a
implication generated from T and N, then we will often denote it by
.
Definition 3.2
[Citation13, Proposition 3.1 and Definition 3.2].
A function is called an
implication if there exist a t-norm T and two fuzzy negations
, N such that
(6)
(6)
Moreover, if I is an
implication generated from T,
and N, then we will often denote it by
.
Definition 3.3
[Citation6, page 232].
Let N be a strict negation, S be a t-conorm and T be a t- norm, such that . Then S is said to be the
dual of T and we denote it by
. In the case that N is a strong negation, then
.
The above Definition 3.3 addresses that a implication generated from a strong negation N is always an
implication and more specific it is
. Thus, the properties of
implications generated from strong negations N are the same with them of
implications, as they are studied in [Citation1, Section 2.4]. So, a
implication generated from a strong negation N always satisfies (NP) and (EP) according to Proposition 2.2. We get the same results for the
implication
(where
is strict) according to the following Proposition 3.1.
Proposition 3.1
Let be an
implication generated from t-norm T and two strict fuzzy negations
, N, such that
. Then
is an
implication. Moreover, it is
.
Proof.
Since is a strict fuzzy negation, then it is a bijection. So there exists the fuzzy negation
. Moreover, if
then for all
it is
Thus
is an
implication. Moreover, it is
Proposition 3.2
[Citation13, Proposition 3.4(i)]
Let be an
implication. Then
.
Remark 3.1
Note that if the corresponding
implication is a
implication. More specific
and
.
3.1. ![](//:0)
, ![](//:0)
Implications and the Neutrality Property (NP)
In this section, we investigate whether or not, and
implications satisfy (NP).
Proposition 3.3
Let be a
implication.
(i) | If N is strong, then | ||||
(ii) | If N is not strong, then |
Proof.
(i) Since N is a strong negation then . Thus
is an
implication and it satisfies (NP) according to Proposition 2.2.
(ii) It is proved in Proposition 3.3(i) in [Citation10].
The above Proposition 3.3 is very important, because it fully characterizes the intersection between the sets of and
implications. This characterization is addressed in [Citation12] Figure .
Proposition 3.4
[Citation13, Proposition 3.4(i)]
Let be an
implication. Then
satisfies (NP), if and only if
.
The above Proposition 3.4 is very important. According to the Lemma 2.1 if , then
is a continuous fuzzy negation,
N is a strictly decreasing fuzzy negation.
Therefore, the following Corollaries are presented without proofs, since they are obvious.
Corollary 3.5
Let be an
implication, where
is not a continuous fuzzy negation. Then
violates (NP).
Corollary 3.6
Let be an
implication, where N is not strictly decreasing. Then
violates (NP).
Corollary 3.7
Let be an
implication, where at least one of
and N is a crisp fuzzy negation. Then
violates (NP).
Proof.
If N is a crisp fuzzy negation, then it is not strictly decreasing and according to the Corollary 3.6, the corresponding violates (NP).
If is a crisp fuzzy negation, then it is not continuous and according to the Corollary 3.5, the corresponding
violates (NP).
Remark 3.2
Corollaries 3.5 and 3.6 are very important. Note that, they prove that if N is not a strictly decreasing fuzzy negation, or
is not a continuous fuzzy negation, then the corresponding
is not an
implication according to Proposition 2.2.
Corollary 3.7 is also very important. Note that, it proves that if at least one of
and N is a crisp fuzzy negation, then the corresponding
is not an
implication according to Proposition 2.2.
Proposition 3.3 can be deduced by Proposition 3.4.
3.2. ![](//:0)
, ![](//:0)
Implications and the Exchange Principle (EP)
In this section, we investigate whether or not, and
implications satisfy (EP).
Proposition 3.8
Let be a
implication.
(i) | If N is strong, then | ||||
(ii) | If N is a crisp fuzzy negation, then | ||||
(iii) | If N is strict, but not strong, then |
Proof.
(i) Since N is a strong negation then . Thus
is an
implication and it satisfies (EP) according to Proposition 2.2.
(ii) It is proved in Theorem 3.2(i) in [Citation10].
(iii) It is proved in Proposition 3.3(ii) in [Citation10].
Proposition 3.9
[Citation13, Proposition 3.5]
Let be an
implication. If
, then
satisfies (EP).
Proposition 3.10
[Citation13, Proposition 3.9(i)]
Let be an
implication, where
and N are crisp fuzzy negations. Then
satisfies (EP).
Proposition 3.11
Let be an
implication. If
satisfies (EP), then
.
Proof.
Since satisfies (EP), then for all
it is
Corollary 3.12
Let be a
implication generated from a non- strong negation N. If
satisfies (EP), then
, for all
.
Proof.
It is deduced by Proposition 3.11, for .
Proposition 3.13
Let be an
implication, where
is strictly decreasing with a fixed point. If N does not have any fixed point, or
and N have different fixed points, then
violates (EP).
Proof.
We assume that satisfies (EP). Therefore, from Proposition 3.11, we deduce that for all
it is
. Since
has a fixed point (this is unique since
is strictly decreasing, therefore it is an injection), there is an
such that
. So,
So,
and N have the same fixed point. That is a contradiction, therefore,
violates (EP).
Remark 3.3
Proposition 3.11 gives us the sufficient condition, that if there is an
such that
, then
violates (EP). This result in the case of
implications, where
is transformed to the sufficient condition, that if there is an
such that
, then
violates (EP).
Note that, if there is an
such that
, then N is not a strong fuzzy negation.
According to [Citation1] Theorem 1.4.7, every continuous fuzzy negation N has a unique fixed point. Therefore, Proposition 3.13 holds if
is strict, N is continuous and they have different fixed points.
Proposition 3.14
Let be an
implication, where
is a strictly decreasing fuzzy negation. The following statements are equivalent:
(i) |
| ||||
(ii) |
|
Proof.
(i)⇒(ii)
Since satisfies (EP), then from Proposition 3.11, we deduce that for all
it is
. Moreover,
is an injection, since it is a strictly decreasing function. So, for any
it is
(ii)⇒(i)
It is deduced by Proposition 3.9.
According to the Lemma 2.1, if , then N is a continuous fuzzy negation. Therefore, by Proposition 3.14, we deduce the following Corollary without proof, since is is obvious.
Corollary 3.15
Let be an
implication, where N is not a continuous fuzzy negation and
is a strictly decreasing fuzzy negation. Then
violates (EP).
A very helpful Theorem is the following.
Theorem 3.16
[Citation1, Theorem 2.4.10]
For a function the following statements are equivalent:
(i) | I is an | ||||
(ii) | I satisfies (I1), (EP) and |
The above Theorem 3.16 helps to the proof of the following Propositions.
Proposition 3.17
Let be an
implication, where
is a continuous fuzzy negation and N is not a strictly decreasing fuzzy negation. Then
violates (EP).
Proof.
We assume that satisfies (EP). By Proposition 3.2 it is deduced that
. Also,
is a continuous fuzzy negation, by the hypothesis. Moreover, since
is a fuzzy implication, it satisfies (I1). So,
satisfies the statement (ii) in Theorem 3.16. By virtue of Theorem 3.16, we deduce that
is an
implication. Thus,
satisfies (NP), according to Proposition 2.2. So, by Proposition 3.4 it must be
. This means that N is strictly decreasing according to Lemma 2.1. That is a contradiction, since N is not strictly decreasing. Thus,
violates (EP).
Proposition 3.18
Let be a
implication generated from a continuous non- strong negation N. Then
violates (EP).
Proof.
We assume that satisfies (EP). By Remark 3.1, it is deduced that
. Also, N is a continuous fuzzy negation, by the hypothesis. Moreover, since
is a fuzzy implication, it satisfies (I1). So,
satisfies the statement (ii) in Theorem 3.16. By virtue of Theorem 3.16, we deduce that
is an
implication. Thus,
satisfies (NP), according to Proposition 2.2. That is a contradiction according to Proposition 3.3(ii), since N is not strong. Thus,
violates (EP).
Corollary 3.19
Let be a
implication generated from a strictly decreasing non- strong negation N. Then
violates (EP).
Proof.
Since N is a non- strong negation, there is an , such that
. We assume that
satisfies (EP), than by virtue of Proposition 3.14 it is
, i.e. N is a strong negation. That is a contradiction. Thus,
violates (EP).
Remark 3.4
Proposition 3.8(iii) can be deduced by Proposition 3.18 or Corollary 3.19.
3.3. ![](//:0)
, ![](//:0)
Implications and the Identity Principle (IP)
In this section, we investigate whether and
implications satisfy (IP).
Proposition 3.20
Let be an
implication. Moreover, let the pair
satisfies (LC). Then
satisfies (IP).
Proof.
For all it is
Corollary 3.21
If is a
implication and the pair
satisfies (LC), then
satisfies (IP).
Proof.
Note that it is Proposition 3.20, where .
Proposition 3.22
Let be an
implication, where
is a non- filling fuzzy negation. Then the following statements are equivalent:
(i) | The pair | ||||
(ii) |
|
Proof.
(i) ⇒ (ii)
It is deduced by Proposition 3.20.
(ii)⇒ (i)
Let satisfies (IP). Then for all
it is
Thus the pair
satisfies (LC).
Corollary 3.23
Let be a
implication generated from a non- filling fuzzy negation N. Then the following statements are equivalent:
(i) | The pair | ||||
(ii) |
|
Proof.
Note that it is Proposition 3.22, where .
In the case, we use a crisp fuzzy negation we get the following implications.
(7)
(7)
(8)
(8)
(9)
(9)
and
(10)
(10)
Proposition 3.24
Let be an
implication, where
is a crisp fuzzy negation. If the pair
satisfies (LC), then
satisfies (IP).
Proof.
Let . For all
, it is
Moreover, the pair
satisfies (LC). So, for all
, it is
Thus,
satisfies (IP).
Let . For all
, it is
Moreover, the pair
satisfies (LC). So, for all
, it is
Thus,
satisfies (IP).
Remark 3.5
(i) It is easy to prove that ,
and
satisfy (IP), if and only if
, since
(11)
(11)
where
,
(12)
(12)
where
and
,
(13)
(13)
where
.
(ii) From (Equation11(I2)
(I2) ) and (Equation13
(I4)
(I4) ), for
we deduce that a
implication generated from a crisp fuzzy negation satisfies (IP).
(iii) It is easy to prove that satisfies (IP), if and only if
, since
(14)
(14)
where
and
.
3.4. ![](//:0)
, ![](//:0)
Implications and the Ordering Property (OP)
In this section, we investigate whether or not, and
implications satisfy (OP). Sometimes (OP) is divided in two sub- properties, the so-called (LOP) and (ROP). It is easy to observe that, if a fuzzy implication I satisfies both (LOP) and (ROP), it satisfies (OP).
Proposition 3.25
[Citation10, Theorem 3.2 and Remark 3.2]
Let be a
implication generated from a crisp fuzzy negation. Then
satisfies (LOP) and violates (ROP) and (OP).
Proposition 3.26
[Citation13, Proposition 3.9(iii),(v)]
Let be an
implication, where
, N are crisp fuzzy negations.
(i) |
| ||||
(ii) |
|
A direct conclusion from Proposition 3.26 is the following.
Corollary 3.27
Let be an
implication, where
, N are crisp fuzzy negations. Then
violates (OP).
Proposition 3.28
Let be an
implication. Moreover, let the pair
satisfies (LC). Then
satisfies (LOP).
Proof.
For all , such that
it is
, since N is decreasing. So,
Thus,
satisfies (LOP).
Proposition 3.29
Let be an
implication generated from strictly decreasing fuzzy negations
, N and a strictly monotone t- norm T such that, the pair
satisfies (LC). Then
satisfies (ROP).
Proof.
For all , such that x>y it is x>0 and
, since N is strictly decreasing. So,
since T is strictly monotone. By virtue of (T1) we have
since
is strictly decreasing. Thus,
satisfies (ROP).
Corollary 3.30
Let be an
implication generated from strictly decreasing fuzzy negations
, N and a strictly monotone t-norm T such that, the pair
satisfies (LC). Then
satisfies (OP).
Proof.
It is deduced by virtue of Propositions 3.28 and 3.29.
Corollary 3.31
Let be a
implication.
(i) | If the pair | ||||
(ii) | If N is strictly decreasing fuzzy negation and the t-norm T is strictly monotone such that, the pair |
Proof.
(i) Note that it is Proposition 3.28 for .
(ii) Note that it is an application of Proposition 3.29 and Corollary 3.30 for .
3.5. ![](//:0)
, ![](//:0)
Implications and the Law of Importation (LI)
In this section, we study whether or not, and
implications satisfy the law of importation (LI) with respect to any, or which, t-norm
.
Proposition 3.32
Let be an
implication. If
, then the couple of functions
and T satisfies (LI). Moreover, this couple of functions is unique, i.e. for any other t-norm
the couple of functions
and
violates (LI).
Proof.
For all , it is
Thus, the couple of functions
and T satisfies (LI).
We assume that, there is an other t-norm , such that the couple of functions
and
satisfies (LI). Then, for all
, it is
So,
Putting z = 0 we have
that is a contradiction, since
.
Corollary 3.33
Let be a
implication generated from a t-norm T and a strong negation N. The couple of functions
and T satisfies (LI). Moreover, this couple of functions is unique, i.e. for any other t-norm
the couple of functions
and
violates (LI).
Proof.
Note that it is Proposition 3.32 since N is strong, therefore .
Corollary 3.34
Let be an
implication, where
is a strict fuzzy negation. Then the couple of functions
and T satisfies (LI). Moreover, this couple of functions is unique.
Proof.
The proof is deduced by Proposition 3.32 since .
Proposition 3.35
Let be an
implication. Let there is an
such that
. Then there is not any t-norm
, such that the couple of functions
and
satisfies (LI).
Proof.
Consider that there is a t-norm , such that the couple of functions
and
satisfies (LI). Then, by Remark 2.4 follows that
satisfies (EP), a contradiction by virtue of Remark 3.3(i).
Corollary 3.36
Let be a
implication generated from a non- strong negation N. Moreover, there is an
such that
. Then there is not any t-norm
, such that the couple of functions
and
satisfies (LI).
Proof.
Note that and apply Proposition 3.35.
Proposition 3.37
Let be an
implication, where
is strictly decreasing with a fixed point. If N does not have any fixed point, or
and N have different fixed points, then there is not any t-norm
, such that the couple of functions
and
satisfies (LI).
Proof.
Consider that there is a t-norm , such that the couple of functions
and
satisfies (LI). Then, by Remark 2.4 follows that
satisfies (EP), a contradiction by virtue of Proposition 3.13.
Proposition 3.38
Let be an
implication, where
is a strictly decreasing fuzzy negation. The following statements are equivalent:
(i) |
| ||||
(ii) | The couple of functions | ||||
(iii) |
|
Proof.
(i)⇔(iii)
See Proposition 3.14.
(ii)⇒(i)
See Remark 2.4.
(iii)⇒(ii)
See Proposition 3.32.
Proposition 3.39
Let be an
implication, where N is not a continuous fuzzy negation and
is a strictly decreasing fuzzy negation. Then there is not any t- norm
, such that the couple of functions
and
satisfies (LI).
Proof.
Consider that there is a t- norm , such that the couple of functions
and
satisfies (LI). Then, by Remark 2.4 follows that
satisfies (EP), a contradiction by virtue of Corollary 3.15.
Proposition 3.40
Let be a
implication generated from a continuous non- strong negation N. Then there is not any t-norm
, such that the couple of functions
and
satisfies (LI).
Proof.
Consider that there is a t- norm , such that the couple of functions
and
satisfies (LI). Then, by Remark 2.4 follows that
satisfies (EP), a contradiction by virtue of Proposition 3.18.
Proposition 3.41
Let be a
implication generated from a strictly decreasing non- strong negation N. Then there is not any t-norm
, such that the couple of functions
and
satisfies (LI).
Proof.
Consider that there is a t-norm , such that the couple of functions
and
satisfies (LI). Then, by Remark 2.4 follows that
satisfies (EP), a contradiction by virtue of Corollary 3.19.
Proposition 3.42
Let be an
implication, generated from a t-norm T and two crisp fuzzy negations
and N. The couple of functions
and
satisfies (LI) if and only if
Proof.
Let . It is
Let
be any t-norm. For all
, it is
and
It is easy to prove that for any t-norm
and
, it is
and
(see [17, Proposition 9]). So we deduce that
(15)
(15)
“⇒”
We assume that the couple of functions and
satisfies (LI). It must be for any
,
“⇐”
We assume that
this means, that
By the above equivalence, we conclude that
Thus, the couple of functions
and
satisfies (LI). If we assume that
. the proof is similar. So it is omitted.
Proposition 3.43
Let be an
implication, generated from a t-norm T and two crisp fuzzy negations
and N. The couple of functions
and
satisfies (LI) if and only if
Proof.
The proof is similar with the proof of Proposition 3.42. So it is omitted.
Remark 3.6
It is obvious that Proposition 3.42 (respectively Proposition 3.43) holds for
implications generated from a t-norm T and a crisp fuzzy negation
(respectively,
).
From Propositions 3.42 and 3.43 we deduce that an
implication (respectively, a
implication) generated from crisp fuzzy negations (respectively negation) always satisfies (LI) with respect to the minimum t- norm
.
From Proposition 3.42 we deduce that
, where N is a crisp fuzzy negation satisfies (LI) with respect to any t-norm
.
From Proposition 3.43 we deduce that
, where N is a crisp fuzzy negation satisfies (LI) with respect to any strictly monotone t-norm
.
Proposition 3.10 can be deduced by Remarks 3.6(ii) and 2.4.
3.6. ![](//:0)
Implications and Φ- Conjugacy Classes
Theorem 3.44
If and
is an
implication, then
is an
implication and moreover
Proof.
Let be an
implication, then
is an
implication according to the Remark 2.3. Moreover, for all
, we deduce that
4. Final Remarks
In this study, we dealt with implications and
implications, which are a generalization of
implications. Firstly, we connected a form of
implications with
implications and we investigated when an
implication satisfies, or not, (NP). In the following, the conditions under an
implication (respectively a
implication) satisfies or violates (EP) have been studied. A study for the satisfaction of (IP) and the satisfaction or violation of (OP) has also been made. Furthermore, except the basic properties of fuzzy implications we expanded our study to the law of importation (LI) with respect to a t-norm
. Our study focused, not only to the satisfaction of violation of (LI) with respect to a t-norm
, but also to the uniqueness, or not of the t-norm
. Moreover, the sufficient and necessary conditions under an
implication (respectively a
implication), generated from crisp fuzzy negations (respectively a crisp fuzzy negation), satisfies (LI) with respect to a t-norm
were presented and proved. Also, the relation of
conjugation in
implications was studied.
Disclosure Statement
No potential conflict of interest was reported by the author(s).
Data Availability Statement
No data were used to support this study.
Additional information
Notes on contributors
Dimitrios S. Grammatikopoulos
Dimitrios S. Grammatikopoulos is with the Department of Civil Engineering, School of Engineering at Democritus University of Thrace, Greece. He received his Bachelor degree in Mathematics in 2005 and his MSc in Statistics and Operational Research in 2008, both from the Department of Mathematics at Aristotle University of Thessaloniki, Greece. He received his second MSc in Applied Mathematics in 2018, from the School of Engineering at Democritus University of Thrace, Greece. His research interests include fuzzy theory and systems with applications. He is currently working towards his PhD degree.
Basil K. Papadopoulos
Basil K. Papadopoulos is currently a Professor at the Department of Civil Engineering, School of Engineering at Democritus University of Thrace, Greece. He is also the Director of the MSc in Applied Mathematics at the School of Engineering at Democritus University of Thrace, Greece. For more information, please visit his webpage: http://utopia.duth.gr/papadob/.
References
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- Grammatikopoulos DS, Papadopoulos BK. An application of classical logic's laws in formulas of fuzzy implications. J Math. 2020;47:777–780. Article ID 8282304. 2020. 18 pages. DOI:https://doi.org/10.1155/2020/8282304.
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