Aggregation of Weak Fuzzy Norms

: Aggregation is a mathematical process consisting in the fusion of a set of values into a unique one and representing them in some sense. Aggregation functions have demonstrated to be very important in many problems related to the fusion of information. This has resulted in the extended use of these functions not only to combine a family of numbers but also a family of certain mathematical structures such as metrics or norms, in the classical context, or indistinguishability operators or fuzzy metrics in the fuzzy context. In this paper, we study and characterize the functions through which we can obtain a single weak fuzzy (quasi-)norm from an arbitrary family of weak fuzzy (quasi-)norms in two different senses: when each weak fuzzy (quasi-)norm is deﬁned on a possibly different vector space or when all of them are deﬁned on the same vector space. We will show that, contrary to the crisp case, weak fuzzy (quasi-)norm aggregation functions are equivalent to fuzzy (quasi-)metric aggregation functions.


Introduction
In mathematics, an aggregation procedure amounts to a method for merging a family of structures of the same type into the only structure of this type. For example, the union or the intersection of subsets of a nonempty set X gave rise to another subset of X by aggregating the family of sets. On the other hand, given a finite family {d i : i = 1, . . . , n} of metrics on X, then max{d 1 , . . . , d n } is also a metric on X that is obtained by merging the original family of metrics. This metric can be constructed by means of the composition of the following functions: • d : X × X → [0, +∞) n given by d(x, y) = (d 1 (x, y), . . . , d n (x, y)); • f : [0, +∞) n → [0, +∞) given by f (x 1 , . . . , x n ) = max{x 1 , . . . , x n }.
In the literature, we can find other schemes of merging mathematical structures. If {(X n , d n ) : n ∈ N} is a countable collection of metric spaces then d(x, y) = ∑ ∞ n=1 min{d n (x n ,y n ),1} 2 n is a metric on ∏ n∈N X n , which is compatible with product topology. In this case, the new metric on ∏ n∈N X n can be obtained with the composition of the following functions: • d : ∏ n∈N X n × ∏ n∈N X n → [0, +∞) N given by d(x, y) = (d n (x n , y n )) n∈N ; In a similar manner, the sup norm · ∞ on R n can be viewed as the aggregation of the absolute value norm on R. This means that this norm is the composition of the following functions: • abs : R n → [0, +∞) n given by abs(x 1 , . . . , x n ) = (|x 1 |, . . . , |x n |); • f : [0, +∞) n → [0, +∞) given by given by f (x 1 , . . . , x n ) = max{x 1 , . . . , x n }.
In all the above cases, a function f is involved in the aggregation process, so it is natural to study which functions allow making these kinds of aggregations. This research

Aggregation of Metrics and Norms
In this section, we compile some results about the aggregation of metrics and norms that constitute a necessary antecedent of our study. We first establish some notations.
We will denote by I an arbitrary index set. The elements of the Cartesian product [0, +∞) I will be written down in bold letters a. Moreover, and for the sake of simplicity, given a ∈ [0, +∞) I , its ith coordinate a(i) will be denoted by a i for any i ∈ I.
We notice that we can endow the set [0, +∞) I with a partial order defined as a b if a i ≤ b i for all i ∈ I. Furthermore, 0 represents the element of [0, +∞) I such that 0 i = 0 for all i ∈ I.
As we have sketched, in the Introduction, that classical constructions of metrics in a Cartesian product are obtained by composing an appropriate function with a Cartesian product of metrics. The study of these functions, called metric preserving functions, has been mainly developed by Borsík and Doboš [1,2]. The corresponding study for quasi-metrics was made by Mayor and Valero [3], who characterized the so-called quasi-metric aggregation functions. In both cases, the underlying idea is to construct a (quasi-)metric in the Cartesian product of a family of (quasi-)metric spaces.
Another related problem was addressed by Pradera and Trillas [6], who studied how to merge a family of pseudometrics defined over the same set into a single one. This question for metrics has been also considered recently by Mayor and Valero [4]. Therefore, we have two different but related problems, which gave rise to two different families of functions that we next define using the terminology of [9,17,20]. 3,4]). A function f : [0, +∞) I → [0, +∞) is called the following: • A (quasi-)metric aggregation function on products if given an arbitrary family of (quasi- A (quasi-)metric aggregation function on sets if given a family of (quasi-)metrics {d i : i ∈ I} on an arbitrary nonempty set X, then f • d is a (quasi-)metric on X where d : for all i ∈ I, x, y ∈ X.
Borsík and Doboš [1] characterized metric aggregation functions on products in the following manner. On its part, Mayor and Valero [3] proved the next result which characterizes the functions merging quasi-metrics on products.
The two previous theorems bring to light that every quasi-metric aggregation function on products is a metric aggregation function on products. However, the reciprocal implication does not hold in general [3] (Example 8).
On the other hand, if you consider (asymmetric) normed vector spaces rather than (quasi-)metric spaces, the concepts of (asymmetric) norm aggregation function on products and (asymmetric) norm aggregation function on sets can be considered in a natural manner. The former has been characterized in [7,8], while the latter has been studied in [9]. As the main objective of this paper is to study this problem in the fuzzy context, we recall the known results for crisp (asymmetric) norms. In the following, a function f : We first recall the following result showing that the family of asymmetric norm aggregation functions on products is equal to the family of norm aggregation functions on products.
From Theorems 1, 2 and 3, we have it that every (asymmetric) norm aggregation function on products is also a (quasi-)metric aggregation function on products. However, the reciprocal implication does not hold in general. We can provide an easy example.
It is straightforward to check that f −1 (0) = 0, and f is subadditive and isotone. Therefore, f is a (quasi-)metric aggregation function on sets. However, f is not positive homogeneous since, for example, f 4 · 1 2 = 1 = 2 = 4 f 1 2 . Hence, f is not an asymmetric norm aggregation function on products.

Weak Fuzzy (Quasi-)Norms
As the goal of the paper is to study in the fuzzy context those functions that aggregate fuzzy norms in the spirit of the results of the previous section, in the following, we present the basic definitions about fuzzy norms and some examples. First, we remind the reader of the well-known notion of a triangular norm. Definition 2 ([21]). We say that a binary operation * : [0, 1] × [0, 1] → [0, 1] is a triangular norm or a t-norm if, for every a, b, c, d ∈ [0, 1], it satisfies the following properties: A t-norm * is said to be continuous if * is a continuous function.

Example 2 ([21]
). Some of the most renowned examples of triangular norms are the following: The origins of fuzzy normed spaces can be found in the concept of probabilistic normed space and Šerstnev space [22,23]. This notion was first adapted to the fuzzy context by Katsaras [24]. Later on, Cheng and Mordeson [25] introduced a new definition of a fuzzy norm, which induces a fuzzy metric in the sense of Kramosil and Michalek [26]. Bag and Samanta considered a more general concept of fuzzy norm [27] by removing the left-continuity condition (see next definition). In this paper, we use the concept of fuzzy norm as considered by Goleţ [19] as well as the terminology of [18,28] relative to (weak) fuzzy (quasi-)norms. Definition 3 ([18,19,28]). A weak fuzzy quasi-norm on a real vector space V is a pair (N, * ) such that * is a continuous t-norm and N is a fuzzy set on V × [0, +∞) such that, for any vectors x, y ∈ V and for any parameters t, s > 0, it satisfies the following conditions: If N also satisfies the following: A (weak) fuzzy norm on a real vector space V is a (weak) fuzzy quasi-norm (N, * ) on V such that the following is the case.

Remark 1.
Notice that the definition of fuzzy norm that we have considered is that of Goleţ. It differs slightly from that defined in [25] by Cheng and Mordeson since they considered a real or complex vector space V. They allow that the parameter t also takes negative values by considering that N(x, t) = 0 for every t < 0, and they only create the definition for the minimum t-norm. The above definition is equal to that given in [18,28].
There are also other notions of a fuzzy norm that modify the previous conditions as the elimination of (FQN5) [27].

Remark 2.
The definition of a (weak) fuzzy (quasi-)normed space (V, N, * ) given above requires the continuity of the triangular norm * . This property is used for ensuring that (N, * ) endows the vector space V with a classical topology τ N (see [18,28]) having as its base, Despite this, since we do not need (N, * ) generating a topology, as we can suppose that * is an arbitrary t-norm rather than a continuous one.
We will use this fact throughout the paper.
We can easily prove that (R, N a , * ) is a fuzzy normed space for every continuous t-norm * .
The reason for having chosen the definition of weak fuzzy (quasi-)norm as considered in 3 instead of other definition of fuzzy norm is that every weak fuzzy (quasi-)norm (N, * ) on a vector space V induces a fuzzy (quasi-)metric (M N , * ) on V (in the sense of the paper [26]) given as M N (x, y, t) = N(y − x, t) for all x, y ∈ V and all t ≥ 0 (see [18]). Since (quasi-)metric aggregation functions have been already characterized in [17] and we plan to characterize here the functions that aggregate fuzzy (quasi-)norms, it is natural to select a concept of fuzzy (quasi-)norm that allows constructing a fuzzy (quasi-)metric.

Aggregation of Weak Fuzzy (Quasi-)Norms
In Section 2 we have summarized the known results about the aggregation of (quasi-)metrics and asymmetric norms on products and on sets. On the other hand, in [17], the functions that aggregate fuzzy (quasi-)metrics on products and on sets were completely characterized. Nevertheless, to the best of our knowledge, the problem for fuzzy (quasi-)norms has not been already solved. The goal of this section is fill in this gap. We first set the definitions of the functions that we intend to characterize. 1] is said to be the following: • A weak fuzzy (quasi-)norm aggregation function on products if given a t-norm * and a collection of weak fuzzy (quasi-)normed spaces{ for every x ∈ ∏ i∈I V i and t ≥ 0.
If the previous condition is only verified for a t-norm * , then F is called an * -weak fuzzy (quasi-)norm aggregation function on products.
• A weak fuzzy (quasi-)norm aggregation function on sets if given a t-norm * and a collection of weak fuzzy (quasi- for every x ∈ V and t ≥ 0. If the previous condition is only verified for a t-norm * , then F is called a * -weak fuzzy (quasi-)norm aggregation function on sets.

Remark 4.
It can be easily proved that a weak fuzzy (quasi-)norm aggregation function on products F : is also a weak fuzzy (quasi-)norm aggregation function on sets. Trivially, if the cardinality of I is one, then the two notions are equivalent. However, if the cardinality of I is greater than one, the two concepts are different in general as we next illustrate with an example.

Example 5.
Consider I an index set and j ∈ I is fixed. Denote by P j the jth projection. It is obvious that P j is a weak fuzzy (quasi-)norm aggregation function on sets In the following, we will introduce several valuable concepts used in [17] for proving a characterization of fuzzy (quasi-)metric aggregation functions which will be also useful in our work. Actually, it will be shown that weak fuzzy (quasi-)norm aggregation functions are exactly the fuzzy (quasi-)metric aggregation functions. Notice that this is not true in the crisp context where norm aggregation functions are metric aggregation functions, but the reciprocal implication does not hold in general [2,3,8,9].
We begin by recalling the notion of (asymmetric) * -triangular triplet [14,17] in which we will use the operation * I on [0, 1] I defined as (a * I b) i = a i * b i for every i ∈ I and every a, b ∈ [0, 1] I . 3 is said to be the following:

Definition 5 ([14,17]). Consider an index set I and a t-norm
in other words, • Asymmetric * -triangular if a * I b c. If (a, b, c) is an asymmetric * -triangular triplet for every t-norm * , then (a, b, c) is called a(n) (asymmetric) triangular triplet. Example 6. Consider a weak fuzzy (quasi-)normed space (V, N, * ). Then (N(x, t), N(y, s), N(x + y, t + s)) is an asymmetric * -triangular triplet for every vector x, y, z ∈ V and every t, s > 0 .
If F preserves * -triangular (asymmetric * -triangular) triplets for every t-norm * , then F is said to preserve triangular (asymmetric triangular) triplets.
We next provide a concept that is a particular case of the concept of domination as considered in [10]. Domination has been demonstrated to be useful in the study of the preservation by means of aggregation functions of some properties of certain fuzzy structures [10,17,30]. It has also been used for constructing other fuzzy structures such as m-polar * -orderings [31]. We will say that F is supmultiplicative if F is * -supmultiplicative for every t-norm * . * -supmultiplicative functions and functions preserving (asymmetric) * -triangular triplets can appear to be unrelated concepts. However, both have been used for characterizing, in different senses, the functions preserving the property of * -transitivity of fuzzy binary relations [10,14]. Its relationship has been disclosed in [17] in the following way. (1) F preserves asymmetric * -triangular triplets; (2) F preserves * -triangular triplets; (3) F is * -supmultiplicative.
If F is isotone then all the above statements are equivalent.
We still need to recall two concepts that will be needed in our characterization.
It is obvious that {s n } n∈N is a nondecreasing sequence and s n t n for all n ∈ N. Moreover, n∈N t n = n∈N s n = t. Suppose that {F(s n )} n∈N converges to F(t) = F( n∈N t n ), that is, n∈N F(s n ) = F( n∈N t n ). Then, the following is the case: since F is isotone. Therefore, n∈N F(t n ) = F( n∈N t n ) = F(t).
Recall (see Section 2) that the characterizations of an asymmetric norm aggregation function or a (quasi-)metric aggregation function f require the imposition of some conditions on the set f −1 (0). These conditions will be substituted by certain properties of the core in the fuzzy framework.

Definition 9 ([17]
). Given a function F : [0, 1] I → [0, 1], its core is the set F −1 (1). We say the following: The core of F is countably included in a unitary face if given {a n : n ∈ N} ⊆ F −1 (1) there exists i ∈ I such that (a n ) i = 1 for all n ∈ N.
In order to obtain a contradiction, suppose that there exists a ∈ [0, 1] I verifying that F(a) = 1 but a = 1. Let J = {i ∈ I : a i = 1}, which is nonempty. Consider the family of weak fuzzy normed spaces {(R, N i , * ) : i ∈ I} where the following is the case. By assumption, (F • N, * ) is a weak fuzzy norm on R I . Let x ∈ R I such that x i = 1 if i ∈ J and x i = 0 otherwise. Then, given t > 0, we have the following: which contradicts (FQN2). Hence, F has trivial core. We next show the isotonicity of F. Consider two elements a, b belonging to [0, 1] I verifying a b. Let us consider the real vector space R and, for every index i ∈ I, we define N i : R × [0, +∞) → [0, 1] as follows.
It is simple to show that (R, N i , * ) is a weak fuzzy normed space for every i ∈ I. Furthermore, N i (1, 2) = a i and N i (1, 3) = b i for every i ∈ I. Since (F • N, * ) is a weak fuzzy norm on R, then F • N(1, ·) is increasing. Consequently, we have the following.
For each i ∈ I, define a function N i : R 2 × [0, +∞) → [0, 1] as follows: where · is the Euclidean norm. Then (N i , * ) is a weak fuzzy norm on R 2 for all i ∈ I. We only verify that (N i , * ) satisfies (FQN4). Let x, y ∈ R 2 and t, s > 0. If x + y = 0, the inequality is trivially true. If x = 0 or y = 0, we also obtain the inequality since N i (z, ·) is isotone for every z ∈ R 2 . Thus, let us suppose that x + y = 0, x = 0 and y = 0. Let j ∈ {1, 2, 3} such that x + y ∈ L j . If x ∈ L j or y ∈ L j , then it is clear that N i (x, t) * N i (y, s) ≤ N i (x + y, t + s). If x ∈ L j and y ∈ L j , we distinguish the following three cases: • If j = 3, then x ∈ L 1 and y ∈ L 2 or viceversa. If t ≤ x or s ≤ y , then N i (x, t) = 0 or N i (y, s) = 0. so the inequality holds. Otherwise t > x and s > y , which implies that t + s > x + y ≥ x + y . Hence, If j = 2, then at least one of x, y belongs to L 3 . Without loss of generality, we can suppose that x ∈ L 3 . As above, if t ≤ x or s ≤ y , then N i (x, t) = 0 or N i (y, s) = 0, so the inequality holds. Otherwise t > x and s > y , which implies that If j = 1, we can reason as in the previous case.
Then, (R, N i , * ) is a weak fuzzy normed space for all i ∈ I. We only check (FQN4). Let x, y ∈ R and t, s > 0. If N i (x, t) = 0 or N i (y, s) = 0, then the conclusion is obvious, so we suppose that N i (x, t) = 0 and N i (y, s) = 0. We may also assume that x = 0, y = 0 and x + y = 0 (otherwise, the conclusion follows trivially). If |x| ≤ t and |y| ≤ s then |x Now suppose that |x| > t and |y| ≤ s. Since N i (x, t) = 0 and x = 0, we also have it that 0 < |x| 2 < t. Then, there exists n x ∈ N such that the following is the case.
Therefore, the following obtains.
If |x| ≤ t and |y| > s, we can reason as above.
Finally, let us suppose that |x| > t and |y| > s. Then, we can find n x , n y ∈ N such that the following is the case.
Then, we have the following.
Thus, F is sequentially left-continuous.
(3) ⇒ (4) We are required to demonstrate that F preserves asymmetric ( * -)triangular triplets. This was proved in [17] (Proposition 3.30), but we will reproduce it here. Let  (a, b, c) ∈ ([0, 1] I ) 3 such that a * I b c. Since F is * -supmultiplicative and isotone, then the following is the case.
be a collection of weak fuzzy quasi-normed spaces. We need to prove that (F • N, * ) is a weak fuzzy quasi-norm on ∏ i∈I V i . Given for all i ∈ I and all t > 0. Since (N i , * ) is a weak fuzzy quasi-norm for all i ∈ I, then Obviously, F • N verifies (FQN3) since given λ, t > 0 and x ∈ ∏ i∈I V i , we the following.
Now, we verify (FQN4). Let x, y ∈ ∏ i∈I V i and t, s > 0. Since (N i , * ) is a weak fuzzy quasi-norm for all i ∈ I, it is clear that the triplet ((N i (x i , t)) i∈I , (N i (y i , s)) i∈I , (N i (x i + y i , t + s)) i∈I ) is asymmetric * -triangular. By hypothesis, (F((N i (x i , t)) i∈I ), F((N i (y i , s)) i∈I ), F((N i (x i + y i , t + s)) i∈I )) is an asymmetric * -triangular triplet, so the following is the case: At last, we must prove (FQN5), that is, F • N(x, ·) is sequentially left-continuous for every x ∈ ∏ i∈I V i . Let {t n } n∈N be a nondecreasing sequence on [0, 1] converging to t. It is clear that { N(x, t n )} n∈N is a nondecreasing sequence on [0, 1] I due to the fact that n∈N converges to N(x, t). Since F is sequentially left-continuous, the conclusion follows.
Observe that, in the crisp context, the norm aggregation functions on products also coincide with the asymmetric norm aggregation functions on products [8,9]. Nevertheless, these functions are different from the (quasi-)metric aggregation functions on products [2,3]. Surprisingly, this does not occur in the fuzzy framework. Proof. This is a direct consequence of the previous theorem and [17] (Theorem 4.15) (notice that, in that paper, sequentially left-continuity is called simply left-continuity).

•
Consider an index set I. Then, the function Inf : [0, 1] I → [0, 1] given by Inf(x) = inf i∈I x i satisfies the conditions of Theorem 6. Therefore, it is a weak fuzzy (quasi-)norm aggregation function on products.
The following theorem, which must be compared with [17] (Theorem 4.19), provides a characterization of the functions that aggregate weak fuzzy (quasi-)norm aggregation on sets.  N(v, 0)) i∈I ) = F(0).
On the other hand, F(1) = F((N(0 V , t)) i∈I ) = F • N(0 V , t) = 1 by (FQN2). For proving that F is * -supmultiplicative, we can proceed as in the proof of this fact in the implication (2) ⇒ (3) of Theorem 6. Now, we check that the core of F is countably included in a unitary face. Suppose, contrary to our claim, that we can find a sequence {a n : n ∈ N} ⊆ F −1 (1) such that for any i ∈ I there exists n i ∈ N verifying (a n i ) i = 1. Let us consider the vector space R and, for each i ∈ I, we define N i : R × [0, +∞) → [0, 1] as the following.
Notice that (R, N i , * ) is a weak fuzzy normed space for all i ∈ I. Let us verify this. It is obvious that (FQN1) is satisfied. On the other hand, let x ∈ R such that N i (x, t) = N i (−x, t) = 1 for all t > 0. By assumption, we can find n i ∈ N such that (a n i ) = 1. Hence, if x = 0 then N i (x, |x| n i ) = (a 1 ) i * . . . * (a n i +1 ) i ≤ (a 1 ) i ∧ . . . ∧ (a n i +1 ) i < 1, which is a contradiction. Therefore x = 0.
We next check (FQN4). Let x, y ∈ R and t, s > 0. If x + y = 0, it is obvious that N i (x, t) * N i (y, s) ≤ N i (x + y, t + s) = 1. Moreover, if x + y = 0 and x = 0 or y = 0, the inequality is also clear since if, for example, y = 0 then N i (x, t) * N i (y, s) = N i (x, t) = (a 1 ) i * . . . * (a n+1 ) i for some n ∈ N. Since t < s + t, the factors that appear in multiplication by the t-norm * in the value of N i (x, t + s) are less or equal than the factors in N i (x, t), so N i (x, t) ≤ N i (x, t + s). Finally, suppose that x + y = 0 and x = 0, y = 0. If t + s > |x + y|, the inequality is clear since N i (x + y, t + s) = (a 1 ) i . Otherwise, t + s ≤ |x + y| ≤ |x| + |y|. Then, t ≤ |x| or s ≤ |y|. We distinguish some of the following cases: • t ≤ |x| and s > |y|. Then, there exists n x ∈ N such that |x| n x +1 < t ≤ |x| n x . Then, the following is the case.
This means that the number of factors that appear in N i (x + y, t + s) is less than or equal to n x + 1, which is the number of factors that appear in N i (x, t). Consequently, t > |x| and s ≤ |y|. In this case, we can reason as above. • t ≤ |x| and s ≤ |y|. Let n x , n y ∈ N such that the following is the case.
Then, we have the following.
This means that the number of factors that appear in N i (x + y, t + s) is less than or equal to max{n x , n y } + 1. By reasoning as above, we obtain the desired inequality. What remains is proving that N i (x, ·) is left-continuous. If x = 0, it is obvious. Suppose now that x = 0 and let t > 0. By construction, if {t n } n∈N is a sequence in (0, +∞) for which its upper limit is t, we can find n 0 ∈ N such that N(x, t n ) is constant for every n ≥ n 0 ; thus, the conclusion follows. We conclude that (N i , * ) is a weak fuzzy norm on R for all i ∈ I.
Notice that if t > 1, then N i (1, t) = (a 1 ) i . Thus, we have the following.
If 0 < t ≤ 1, then we can find n ∈ N such that N i (1, t) = (a 1 ) i * . . . * (a n+1 ) i for all i ∈ I. Since F is * -supmultiplicative, then we have the following.
Therefore, F • N(1, t) = 1 for all t > 0, which contradicts the fact that (F • N, * ) is a weak fuzzy norm. Consequently, the core of F is countably included in a unitary face.
The proofs that F is isotone and the proof that F is sequentially left-continuous are similar to the same proofs in the implication (2)⇒ (3) of Theorem 6.
Hence, F((N i (x, t) * N i (−x, t)) i∈I ) = 1 for all t > 0. Let us define a n = (N i (x, 1 n ) * N i (−x, 1 n )) i∈I . Then, {a n : n ∈ N} ⊆ F −1 (1). Since the core of F is countably included in a unitary face, then (a n ) j = 1 for some j ∈ I and for all n ∈ N, that is, N j (x, 1 n ) * N j (−x, 1 n ) = 1 for all n ∈ N. Consequently, N j (x, 1 n ) = N j (−x, 1 n ) = 1 for all n ∈ N. Moreover, since N i (x, ·) and N i (−x, ·) are isotone, we immediately obtain that N j (x, t) = N j (−x, t) = 1 for all t > 0. Since (N i , * ) is a weak fuzzy quasi-metric on V, then x = 0 V . Therefore, F • N satisfies (FQN2).
It is clear that F • N verifies (FQN3) since, given λ, t > 0 and x ∈ V, we have the following.
In order to prove (FQN4), let x, y ∈ V and t, s > 0. Since (N i , * ) is a weak fuzzy quasinorm for all i ∈ I, it is obvious that ((N i (x, t)) i∈I , (N i (y, s)) i∈I , (N i (x + y, t + s)) i∈I ) is an asymmetric * -triangular triplet. By hypothesis, (F((N i (x, t)) i∈I ), F((N i (y, s)) i∈I ), F((N i (x + y, t + s)) i∈I )) is an asymmetric * -triangular triplet. Thus, the following is the case.
F((N i (x, t)) i∈I ) * F((N i (y, s)) i∈I ) ≤ F((N i (x + y, t + s)) i∈I ) Thus, F • N verifies (FQN4). To this end, (FQN5) follows in a similar manner as in the implication (4)⇒ (1) of Theorem 6. Proof. This is a direct consequence of the previous theorem and [17] (Theorem 4.19) (notice that in that, in the paper, sequentially left-continuity is called simply left-continuity).
We provide an example of a ∧-weak fuzzy (quasi-)norm aggregation function on sets that are not a ∧-weak fuzzy (quasi-)norm aggregation function on products.
Furthermore, if F(a) = 1, then a min I = 1. Otherwise, min I · a min I < min I, which implies that inf{i · x i : i ∈ I} ≤ min I · a min I < min I. Thus, F(a) < 1 follows, which is a contradiction. Consequently, the core of F is countably included in a unitary face. Thus, F is a ∧-weak fuzzy (quasi-)norm aggregation functions on sets.
However, the core of F is not trivial. In fact, given j ∈ I\{min I}, then F(b) = 1 where b ∈ [0, 1] I is given by b i = 1 whenever i = j and b j = min I j . Author