1 Introduction
The concept of the fuzzy set was introduced by Zadeh (1965) and, since then, several mathematical concepts such as number, group, topology, differential equation, and so on, have been fuzzified. There are several ways to extend the propositional connectives for a set , but in general these extensions do not preserves all the properties of the classical logical connectives. Triangular norms (tnorms) and triangular conorms (tconorms) were first studied by Menger [menger] and also by Schweizer and Sklar [sklar] in probabilistic metric spaces and they are used to represent the logical conjunction in fuzzy logic and the interception of fuzzy sets, whereas tconorms are used to represent the logical disjunction in fuzzy logic and the union in fuzzy set theory.
In 1965, L. A. Zadeh introduced the notion of fuzzy negation in [Zadeh65], known as standard negation, in order to represent the logical negation and the complement of fuzzy sets. Since then, several important classes of fuzzy negations have been proposed with different motivations, as we can see in [est, FKL99, hig, lowen, ov, trillas]. Fuzzy negations have applications in several areas, such as decision making, stock investment, computing with words, mathematical morphology and associative memory, as the presented in [bedregal12, bent, FKL99, GMM16, VS08, zhao].
The ordinal sums construction was first introduced, in the context of semigroups, by Climescu in [Cli46] and Clifford in [Cli54]. In the context of fuzzy logic, the ordinal sums were first studied for triangular norms and triangular conorms in [SS63] in order to provide a method to construct new tnorms and tconorms from other tnorms and tconorms (for more details see [klement]). However, the ordinal sums of several others important fuzzy connectives also has been studied, such as, for example, the ordinal sums of copulas [Nel06], overlap functions [DB14], uninorms [MeZ16, MeZ17], fuzzy implications [DK16, Su15] and fuzzy negations [BSJIFS]. In particular, the ordinal sums of fuzzy negations proposed in [BSJIFS] were made in the context of Morgan’s triples and they were not deeply studied.
In this paper, we consider the notion of the ordinal sums of a family of fuzzy negations, as introduced in [BSJIFS], and prove some results involving these concepts. In particular, we establish conditions for the ordinal sum of a family of fuzzy negations resulting in a fuzzy negation belonging to a class of fuzzy negations, such as strict, strong, frontier, continuous and invertible.
This paper is organized as follows: Section 2 provides a review of concepts such as tnorms, tconorms, fuzzy implications, fuzzy negations, natural fuzzy negations, ordinal sums of a family of tnorms, tconorms and fuzzy implications. In Section 3, we prove that the ordinal sum of a family of fuzzy negations is a fuzzy negation and we prove results involving concepts of ordinal sums of a family of fuzzy negations and equilibrium point. In Section 4, we establish conditions for the ordinal sum of a family of fuzzy negations resulting in a fuzzy negation belonging to a class of fuzzy negations such as strict, strong, frontier, continuous and invertible. In Section 5, we define the left ordinal sum of a family of fuzzy implications and prove results involving ordinal sums of a family of tnorms, tconorms and fuzzy implications. Also, we prove that the natural negation of the left ordinal sum of a family of fuzzy implications is the same to the ordinal sum of a family of fuzzy negations. Finally, Section 6 contains the final considerations and future works.
2 Preliminaries
In this section, we will briefly review some basic concepts which are necessary for the development of this paper. The definitions and additional results can be found in [alsina, atanassov99, bedregal10, bedregal12, bustince99, bustince03, FodorRoubens1994, klement].
2.1 tnorms, tconorms, fuzzy implications and fuzzy negations
Definition 2.1
A function is a tnorm if, for all , the following axioms are satisfied:
1. Symmetry: ;
2. Associativity: ;
3. Monotonicity: If , then ;
4. One identity: .
A tnorm is called positive if it satifies the condition: iff or .
Example 2.1
Some examples of tnorms:
1. Gödel tnorm: ;
2. Product tnorm: ;
3. Łukasiewicz tnorm: ;
4. Drastic tnorm:
Definition 2.2
A function is a tconorm if, for all , the following axioms are satisfied:
1. Symmetry: ;
2. Associativity: ;
3. Monotonicity: If , then ;
4. Zero identity: .
A tconorm is called positive if it satifies the condition: iff or .
Example 2.2
Some examples of tconorms:
1. Gödel tconorm: ;
2. Probabilistic sum: ;
3. Łukasiewicz tconorm: ;
4. Drastic sum:
Definition 2.3
[BJ08, Definition 1.1.1] A function is called a fuzzy implication if it is satisfies the following conditions:
J1: is nonincreasing with respect to the first variable;
J2: is nondecreasing with respect to the second variable;
J3: ;
J4: ;
J5: .
Example 2.3
Some examples of fuzzy implications:
1. Gödel implication:
2. Rescher implication:
3. KleeneDienes implication: .
A function is a fuzzy negation if
N1: and ;
N2: If , then , for all .
A fuzzy negations is strict if it is continuous and strictly decreasing, i.e., when . A fuzzy negation satisfying the condition N3 is called strong.
N3: for each .
A fuzzy negation is called crisp if it satisfies N4
N4: For all .
Example 2.4
Some examples of fuzzy negations:
1. Standard negation: ;
2. A strict nonstrong negation: ;
3. A strong nonstandard negation: .
Note that:

If is strong then it has an inverse which is also a strict fuzzy negation;

If is strong then is strict.
A fuzzy negation is said to be nonvanishing if iff and is said to be nonfilling if iff . A fuzzy negation that is simultaneously nonvanishing and nonfilling is called frontier [DB14b].
An equilibrium point of a fuzzy negation is a value such that .
Definition 2.4
[BJ08, Definition 2.3.1] Let be a tnorm. The function defined as
for each , is called the natural fuzzy negation of or the negation induced by . In addition, let be a tconorm. The function defined as
for each , is called the natural fuzzy negation of or the negation induced by .
Remark 2.1
[BJ08, Remark 2.3.2 (i)]
Clearly and are, in fact, fuzzy negations.
Definition 2.5
[BJ08, Definition 1.4.15] Let be a fuzzy implication. The function defined by
(1) 
for all , is called the natural negation of or the negation induced by .
Remark 2.2
[BJ08, Lemma 1.4.14]
Clearly is in fact a fuzzy implications.
Definition 2.6
[BSJIFS, Definition 2.5]
Let be a tnorm, be a tconorm and be a strict fuzzy negation. is the dual of if, for all , . Similarly, is the dual of if, for all , .
Proposition 2.1
[weber, Theorem 3.2]
Let be a tnorm, be a tconorm and be a strict fuzzy negation. Then, is a tconorm and is a tnorm.
If the negation is standard, then is called dual tconorm of and is called dual tnorm of .
2.2 Ordinal sums of tnorms, tconorms and fuzzy implications
In this subsection, we will introduce the notion of ordinal sums of a family of tnorms and tconorms, and some important results that will be used in the course of this work. For more information, see [BSJIFS, klement, weber].
Proposition 2.2
[klement] Let be a family of tnorms and be a family of nonempty, pairwise disjoint open subintervals of . Then the function defined by
(2) 
is a tnorm which is called the ordinal sum of the summands .
Proposition 2.3
[klement] Let be a family of tconorms and be a family of nonempty, pairwise disjoint open subintervals of . Then the function defined by
(3) 
is a tconorm which is called the ordinal sum of the summands .
Nevertheless, for fuzzy implications there are several proposal of ordinal sums. For example,
Proposition 2.4
[DK161, Theorem 7] Let be a family of implications and be a family of nonempty pairwise disjoint open subintervals of such that for each . Then the function given by
(4) 
is an implication which is called the ordinal sum of the summands .
Other proposal of ordinal sums for fuzzy implications can be found, for example, in [B17, DK16, DK161, DK17, Su15].
3 Ordinal sums of fuzzy negations
In this section, we will use the definition of ordinal sums of fuzzy negations introduced in [BSJIFS], to show some results involving equilibrium point, relations between some classes of fuzzy negations and that ordinal sum of a fuzzy negation family is a fuzzy negation.
Definition 3.1
[BSJIFS, Definition 3.1] Let be a family of fuzzy negations and be a family of nonempty, pairwise disjoint open subintervals of . Then the function defined by
(5) 
is called of the ordinal sum of the summands .
Lemma 3.1
[BSJIFS, Lemma 3.1] Let be a family of nonempty, pairwise disjoint open subintervals of , be a family of fuzzy negations and the ordinal sum of the summands . Then,
1) If for some , then ;
2) If , then .
Proposition 3.1
[BSJIFS, Proposition 3.1]
Let be a family of nonempty, pairwise disjoint open subintervals of and be a family of fuzzy negations.
Then the ordinal sum of the summands is a fuzzy negation.
If is a family of fuzzy negations such that is also a family of nonempty,
pairwise disjoint open subintervals of , then the ordinal sum of and with respect to
will be denoted by and , respectively.
Proposition 3.2
Let be a family of nonempty, pairwise disjoint open subintervals of , be a family of fuzzy negations and be the ordinal sum of the summands . If, for some , has an equilibrium point and , then is the equibibrium point of .
Proof: Suppose that , for all . Then, and therefore . Since, and , then
Therefore, is the equilibrium point of .
Proposition 3.3
Let be a family of nonempty, pairwise disjoint open subintervals of , be a family of fuzzy negations and be the ordinal sum of the summands . Then if and only if for all .
Proof: () Let and . Then . So,
() Suppose that for all . Then
Proposition 3.4
Let be a family of nonempty, pairwise disjoint open subintervals of , be a family of fuzzy negations and be the ordinal sum be the summands . Then, if and only if for all .
Proof: Analogous from Proposition 3.3.
4 Ordinal sums of fuzzy negations and classes of fuzzy negations
In this section, we will prove some propositions and theorems using definitions and results introduced in the previous sections. We will establish conditions for the ordinal sum of a family of fuzzy negations resulting in a fuzzy negation belonging to a class of fuzzy negations such as strict, strong, frontier, continuous and invertible.
Proposition 4.1
Let be a family of nonempty, pairwise disjoint open subintervals of and be a family of functions and the function obtained as in Eq. (8). All the ’s are (continuous, strictly decreasing) fuzzy negations if and only if then is a (continuous, strictly decreasing) fuzzy negation such that and for each .
Proof: () If all the ’s are fuzzy negations, then, by Proposition 3.1, is a fuzzy negation. In addition, for each , .
Now, suppose that for each , is continuous. Then, is clearly continuous. Since, is continuous then it is sufficient to prove that for each , and . In fact,
Analogously, we prove that .
Now we will prove that is strictly decreasing when all are strictly decreasing. If then we have the following cases:
 Case 1:

If for some , then and therefore . So, by Eq. (8), .
 Case 2:

If and for some such that then . So, by Lemma 3.1, and . Thus, since , then .
 Case 3:
 Case 4:
 Case 5:

If then by Eq. (8), .
Therefore, is strictly decreasing.
() If is a fuzzy negation such that and then, for each ,
Analogously,
Let , such that and and . Then, and therefore,
. So, . Hence, .
Therefore, is a fuzzy negation for each .
In addition, from Eq. (8), clearly for each , if is continuous then also is continuous and if
is strictly then also is strict.
Proposition 4.2
Let be a family of nonempty, pairwise disjoint open subintervals of and be a family of fuzzy negations such that is nonfilling when and nonvanishing when . Then, the ordinal sum of the summands is frontier.
Proof: By Proposition 3.1, is a fuzzy negation. Let . If
then, from Eq. (8), . Suppose that for some . If and
then, by Lemma 3.1,
, i.e. . If then . So,
because is nonfilling, . Analogously, if then .
So, because is nonvanishing, . Therefore, for each , , i.e.
is frontier.
Proposition 4.3
Let be a family of nonempty, pairwise disjoint open subintervals of and be a family of fuzzy negations. If the ordinal sum of the summands satisfies the following two properties

for some only when ; and

for some only when
then is frontier for each .
Proof: Suppose that for some , is not frontier. Then there exists such that or
there exists such that . Let and therefore, . In the first case, we have that
. So, by the second property, and therefore, which is a contradiction.
The second case is analogous.
Therefore, for each , the fuzzy negation is frontier.
Theorem 4.1
Let be a family of nonempty, pairwise disjoint open subintervals of and be a family of fuzzy negations. If all the ’s are strict fuzzy negations and, for each , there exists such that and , then the ordinal sum of the summands , is a strong fuzzy negation.
Proof: From Proposition 3.1, is a fuzzy negation. Besides this, for any if for some then for hypothesis there exists such that and by Eq. (8), . Therefore,
If then and, by Lemma 3.1 and hypothesis,
.
So, .
Theorem 4.2
Let be a family of nonempty, pairwise disjoint open subintervals of and be a family of fuzzy negations. If the ordinal sum of the summands , is a strong fuzzy negation then all the ’s are strict fuzzy negations and, for each , there exists such that and . In addition, if for each ,
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