Zero-sets of copulas

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Zero-sets of copulas Enrique de Amo a,∗ , Juan Fernández-Sánchez b , Manuel Úbeda-Flores a a Department of Mathematics, University of Almería, 04120 Almería, Spain b Research Group of Mathematical Analysis, University of Almería, 04120 Almería, Spain

Received 13 April 2019; received in revised form 10 September 2019; accepted 17 September 2019

Abstract We study conditions on sets in order to be zero-sets of semi-copulas, quasi-copulas, and with special attention, of copulas. We find necessary and sufficient conditions for characterizing the zero-sets of absolutely continuous copulas and copulas whose support coincides with the closure of the complementary of the zero-set. Moreover, we study several topological properties and the lattice-theoretic structure, and characterize the zero-sets of the class of Archimedean copulas. © 2019 Published by Elsevier B.V. Keywords: Archimedean copula; Copula; Quasi-copula; Semi-copula; Zero-set

1. Introduction Aggregation functions play an important role in many applications of fuzzy set theory or fuzzy logic, among many other fields, since they are used to convert finitely many input values into a single representative output value (see, e.g. [5,28]). Special subclasses of aggregation functions are semi-copulas, t-norms, quasi-copulas and copulas. These functions are called conjunctors, because they extend the classical Boolean conjunction, and hence, the zero-set of a binary aggregation function is of particular interest. of an aggregation function We recall that the zero-set   A, denoted by ZA , is the inverse image of the value 0, i.e. ZA := A−1 (0) = (x, y) ∈ [0, 1]2 : A(x, y) = 0 , and the boundary curve of ZA is the set B (ZA ) = {(x1 , y1 ) ∈ ZA :  (x2 , y2 ) ∈ ZA such that x1 < x2 and y1 < y2 }. Several studies of zero-sets can be found in the literature: for example, the boundary curve of the zero-set of left-continuous t-norms is formed by a strong negation in [35]. A continuous t-norm T is Archimedean if, and only if, there exists a continuous and strictly decreasing function f from [0, 1] to [0, +∞] with f (1) = 0 such that T (x, y) = f (−1) (f (x) + f (y)) (see [34]), where (−1) is the quasi-inverse of f , i.e. f (−1) (x) = sup{t ∈ [0, 1] : f (t) ≤ x}. If f (0) = +∞, then we have Z = f T   (x, y) ∈ [0, 1]2 : y ≤ f (−1) (f (0) − f (x)) . * Corresponding author.

E-mail addresses: [email protected] (E. de Amo), [email protected] (J. Fernández-Sánchez), [email protected] (M. Úbeda-Flores). https://doi.org/10.1016/j.fss.2019.09.012 0165-0114/© 2019 Published by Elsevier B.V.

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Let z = T (x, y) be the graph of a continuous t-norm T over [0, 1]2 , and suppose that the remainder of the surface consists of the line segments joining the points on B(ZT ) to the point (1, 1, 1). This construction is known as conic t-norm [1], and constitutes members of the family of Yager t -norms [50]. Inspired by this idea, conic aggregation functions, including semi-copulas, quasi-copulas and copulas, are studied in [31], where the authors characterize (for the n-dimensional case) the zero-set of a conic aggregation function. Few studies on zero-sets of aggregation functions with restrictions are found in the literature. In the case of aggregation functions without restrictions, finding conditions is immediate (see, e.g. [30]). Our aim in this paper is to study conditions such that a set can be the zero-set of certain special types of (bivariate) aggregation functions. After some preliminaries (Section 2), in Section 3 we study conditions on sets to be zero-sets of semi-copulas, quasi-copulas or copulas, and where we also study several topological properties and the lattice-theoretic structure, and pay special attention to the case of Archimedean copulas. Conclusions are provided in Section 4. 2. Preliminaries A (binary) aggregation function A : [0, 1]2 −→ [0, 1] is a non-decreasing function (in each component) such that A(0, 0) = 0 and A(1, 1) = 1. An aggregation function A is a semi-copula if A(x, 1) = A(1, x) = x for all x in [0, 1] (see [13]). This concept was introduced in order to investigate some notions of bivariate ageing for pairs of exchangeable random variables [4]. A semi-copula T is a triangular norm (t-norm for short) if T is both commutative, i.e. T (x, y) = T (y, x) ∀ x, y ∈ [0, 1], and associative, i.e. T (T (x, y), z) = T (x, T (y, z)) ∀ x, y, z ∈ [0, 1] (see [33]). Triangular norms appeared first in the context of probabilistic metric spaces [38,44], and later on, in fuzzy set theory [3]. A semi-copula S is a quasi-copula if it satisfies the 1-Lipschitz condition, i.e. |S(x, y) − S(x , y )| ≤ |x − x | + |y − y | ∀ x, x , y, y ∈ [0, 1] (see [27]). They were introduced in [2] in order to characterize operations on distribution functions that can, or cannot, be derived from operations on random variables defined on the same probability space. For a complete survey on quasi-copulas, we refer to [45]. A semi-copula S is a copula if it satisfies the 2-increasing property, i.e. S(x , y ) − S(x , y) − S(x, y ) + S(x, y) ≥ 0 ∀ x, x , y, y ∈ [0, 1] such that x ≤ x and y ≤ y . We denote by C the set of all copulas. Note that any copula is a quasi-copula, since the 2-increasingness of a semi-copula implies its 1-Lipschitz continuity. Note also that a commutative and associative copula is a t-norm; and a t-norm which satisfies the 1-Lipschitz condition is a copula. The importance of copulas in probability and statistics comes from Sklar’s theorem [47], which shows that the joint distribution function H of a pair of random variables and the corresponding marginal distributions F and G are linked by a copula C in the following manner: H (x, y) = C(F (x), G(y)),

∀ (x, y) ∈ [−∞, ∞]2 .

If F and G are continuous, then the copula is unique; otherwise, the copula is uniquely determined on Range F × Range G (see [10] for details). For a complete and constructive proof of Sklar’s theorem, see [49], and for a complete survey on copulas, we refer to [14,40]. When Q is a quasi-copula, but it is not a copula, then Q is called a proper quasi-copula. Let  denote the copula of two independent random variables, i.e. (x, y) = xy ∀ (x, y) ∈ [0, 1]2 . Every (quasi-)copula Q satisfies the following inequalities: W (x, y) = max(0, x + y − 1) ≤ Q(x, y) ≤ min(x, y) = M(x, y),

∀ (x, y) ∈ [0, 1]2 .

Note that W and M are copulas themselves. We need some notions of measure theory (see e.g. [29]). Let B([0, 1]) and B([0, 1]2 ) denote the respective Borel σ -algebras in [0, 1] and [0, 1]2 , and λ denotes the Lebesgue measure on [0, 1]. A measure μ on B([0, 1]2 ) is doubly stochastic if μ(B ×[0, 1]) = μ([0, 1] ×B) = λ(B) for every B in B([0, 1]). Each copula C induces a doubly stochastic measure μC by setting μC (R) = VC (R) for every rectangle R ⊆ [0, 1]2 (where VC (R) is defined as C(x , y ) − C(x , y) − C(x, y ) + C(x, y) for x, x , y, y in [0, 1] such that x ≤ x and y ≤ y , being (x , y ), (x , y), (x, y ) and (x, y) the vertices of R), and extending μC to the σ -algebra B([0, 1]2 ). The measure μC is the sum of an absolutely ac continuous measure μac C with respect to the 2-dimensional Lebesgue measure λ2 (i.e. λ2 (B) = 0 implies μC (B) = 0   for B ∈ B [0, 1]2 ) and a singular measure μsC (i.e. the measure is concentrated on a set B such that λ2 (B) = 0).   Thus, for any copula C such that μsC [0, 1]2 = α > 0, C has a singular component of total mass equal to α. A copula

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C is said to be singular (respectively, absolutely continuous) if the measure μC induced by C is singular (respectively, absolutely continuous) with respect to λ2 (see [15,17] for more details). Note that the copulas W and M are singular, while the independence or product copula  is absolutely continuous. The support of a copula C is the complement of the union of all open subsets of [0, 1]2 with μC -measure that equals zero. When we refer to the “mass” on a set, we mean the value of μC for that set. 3. Zero-sets of special types of aggregation functions Finding conditions for a set to be the zero-set of an aggregation function without restriction is immediate. In this section we wonder about conditions on sets to be the zero-sets of special types of aggregation functions. 3.1. Zero-sets of semi-copulas We have the following result that characterizes zero-sets for semi-copulas. Theorem 1. A set Z is the zero-set of a given semi-copula if, and only if, {(0, 1), (1, 0)} ⊂ Z and it is satisfied that if (x, y) ∈ Z, then [0, x] × [0, y] ⊆ Z. Proof. Suppose Z is the zero-set of a semi-copula. Boundary conditions for semi-copulas guarantee that {(0, 1), (1, 0)} ⊂ Z and monotonicity in each variable leads us to the fact that if (x, y) ∈ Z then we have [0, x] × [0, y] ⊆ Z. Conversely, for the set Z, consider the function  0, if (x, y) ∈ Z, S(x, y) = M(x, y), otherwise. It is easy to prove that S is a semi-copula (we omit its proof), whence the result follows. 2 3.2. Zero-sets of quasi-copulas and copulas Unlike the case of semi-copulas, there exists a continuity framework for quasi-copulas and copulas, and therefore the supremum of the inverse image of a compact interval by a continuous function is, in fact, a maximum. Moreover, the results in this subsection are valid for both quasi-copulas and copulas too (because continuity properties), whereby we will focus our study on the case of copulas. We begin with the following definition (see, e.g. [5,22]). Definition 1. A function η : [0, 1] −→ [0, 1] is said to be a negation if it is a decreasing monotone function such that η(0) = 1 and η(1) = 0. A negation η : [0, 1] −→ [0, 1] is called a c-negation if it is a left-continuous negation such that η(x) ≤ 1 − x ∀ x ∈ [0, 1]. For each copula C, let ηC : [0, 1] −→ [0, 1] be the function defined by ηC (x) := max{y ∈ [0, 1] : C(x, y) = 0}.

(1)

Then we have the following result. Proposition 2. Given a copula C, the function ηC defined by (1) is a c-negation. Proof. Since each copula C is continuous and non-decreasing in each variable, it is clear that ηC is a left-continuous and decreasing function satisfying ηC (0) = 1 and ηC (1) = 0. Moreover, since max(0, x + y − 1) ≤ C(x, y) ∀ (x, y) ∈ [0, 1]2 , we have that ηC (x) ≤ 1 − x ∀ x ∈ [0, 1]. 2 Remark 1. We want to stress that in the Proof of Proposition 2 we have not used the 2-increasing property, so that the result is also true for quasi-copulas. In general, results concerning copulas and quasi-copulas in relation to zero-sets are not identical. For example, recall the sets of conic copulas and conic quasi-copulas discussed in [31] (we will

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Fig. 1. Mass distribution of the sparse copula Cx,y .

mention these functions later). Conic copulas are characterized by zero-sets related to convex c-negations, while for quasi-copulas one can consider a wider family of c-negations. We now provide the main result of this subsection. Theorem 3. For each c-negation f there exists a copula C such that f = ηC . Proof. For (z, t) ∈ [0, 1]2 , with t < 1 − z, we denote by Cz,t the sparse copula [12,14] defined by (a) J11 = [0, z], J21 = [z, 1 − t], J31 = [1 − t, 1], J32 = [0, t], J22 = [t, 1 − z], J12 = [1 − z, 1]; (b) C1 = C3 = W and C2 is a copula satisfying C2 (x, y) > 0 when xy > 0 (see Fig. 1). The expression for the copula Cz,t is given by





 3   x − an1 y − an2 1   ,g   Cz,t (x, y) = λ Jn Cn g , (2) λ Jn1 λ Jn1 n=1

where ani , bni = Jni , for i = 1, 2, n = 1, 2, 3, and g(t) = max(0, min(t, 1)). For a c-negation f , since f (z) ≤ 1 − z for every z in [0, 1], consider the copula Cz,f (z) , which we denote by Cz . If {an }n∈N is a dense sequence in ]0, 1[ then C(x, y) :=

 1 Ca (x, y) 2n n

n∈N

is a copula such that f = ηC . To see this, for (x, y) in [0, 1]2 such that y ≤ 1 − x, we consider two cases: 1. Let (x, y) be such that y ≤ f (x), and suppose Can (x, y) > 0. We have two subcases: i. an < x < 1 − f (an ). In this case we have f (x) ≤ f (an ). ii. f (an ) < y < 1 − an . From i. and ii. we have f (x) < y, which is absurd, so Can (x, y) = 0 and C(x, y) = 0. 2. If (x, y) is a point such that y > f (x), we define z0 := inf{z ∈ [0, 1] : f (z) < x + y − z}. We have z0 < x and f (z0 ) < y, so that there exists εx,y > 0, depending on (x, y), such that if an ∈]z0 , z0 + εx,y [, we have (x, y) ∈ ]an , 1 − f (an )[×]f (an ), 1 − an [, and hence Can (x, y) > 0. Therefore, C(x, y) ≥

Can (x, y) > 0. 2n

This completes the proof.

2

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Remark 2. We want to stress that another proof of Theorem 3 is possible if we use the functions gx,y (z) = Cz (x, y), where Cz denotes the copulas used in the proof of the theorem above. After proving that they are in fact measurable functions, we define 1 E(x, y) :=

gx,y (z) dz. 0

Such E is a copula, and the equality f = ηE is also satisfied. Corollary 4. Given a set Z ⊂ [0, 1]2 , there exists C ∈ C such that Z = ZC if, and only if, the following four conditions hold: (a) (b) (c) (d)

Z is closed; Zx = {y ∈ [0, 1] : (x, y) ∈ Z} = ∅; if (x, y1 ) ∈ Z, then (x, y) ∈ Z for all y ∈ [0, y1 ]; the function f (x) = max Zx is a c-negation.

If we take C2 = M in the Proof of Theorem 3, we obtain the following result that summarizes proposition and theorem above. Corollary 5. A function f is a c-negation if, and only if, there exists a singular copula C such that ηC = f . As a consequence of Corollary 5, a natural question arises: Is it possible to find an absolutely continuous copula with the same property as the singular copula given in Corollary 5? We provide an answer in the next result. Before that, we need to recall some previous concepts. Let {Ji } = {[ai , bi ]}, ai < bi , be a collection of subintervals of [0, 1] with disjoint interiors. Let {Ci } be a collection of copulas with the same indexing as {Ji }. Then, the ordinal sum of the collection {Ci } with respect to {Ji } is the copula C defined by (see [14,40]) ⎧   ⎪ ⎨ ai + (bi − ai ) · Ci u − ai , v − ai , (u, v) ∈ J 2 , i b i − a i b i − ai C(u, v) = ⎪ ⎩ M(u, v), otherwise. Note that the ordinal sum of copulas is related to the main diagonal section of [0, 1]2 and it is based on the copula M. A similar construction can be considered for the opposite diagonal section based on the copula W , the so-called W -ordinal sum (see [39]). The definition (and properties) concerning W -ordinal sum of copulas can be obtained from the standard ordinal sum of copulas by using the fact that for any copula C, the transformations C1(x, y) = x − C(x, 1 − y) and C2 (x, y) = y − C(1 − x, y) are also copulas (see [40]). We will use the so-called rectangular patchwork construction for copulas, which consists of constructions based on the redefinition of a known copula on some rectangles in the unit square (see [21]). In the sequel, for any bijection h : [0, 1] −→ [0, 1], with h−n , n ∈ N, we will denote compositions of the inverse of h with itself, i.e. h−1 , n times. Proposition 6. For a c-negation f , there exists an absolutely continuous copula C such that f = ηC if, and only if, λ ({x ∈ [0, 1] : f (x) = 1 − x}) = 0. Proof. First suppose there exists an absolutely continuous copula C such that f = ηC . If we have {x ∈ [0, 1] : f (x) = 1 − x} = {0, 1}, then the measure is zero. Otherwise, C is a W -ordinal sum. The measure μC of the W -ordinal sum holds μC ({(x, 1 − x) : x ∈ [0, 1]}) = λ ({x ∈ [0, 1] : C (x, 1 − x) = 0}). Since C is absolutely continuous, then we have λ ({x ∈ [0, 1] : C (x, 1 − x) = 0}) = 0. Moreover, since C (x, 1 − x) = 0 if, and only if, f (x) = 1 − x, we conclude λ ({x ∈ [0, 1] : f (x) = 1 − x}) = 0. Conversely, if λ ({x ∈ [0, 1] : f (x) = 1 − x}) = 0, the existence of an absolutely continuous copula C such that ηC = f is equivalent to the existence of an absolutely continuous copula associated with the interval [a, b] such that f (a) = 1 − a, f (b) = 1 − b and f (z) < 1 − z with z ∈]a, b[. Specifically, we need a copula for each function

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Fig. 2. Example of case 1. in the Proof of Proposition 6.

fa,b (x) =

f (a + (b − a)x) − f (b) f (a) − f (b)

since fa,b is a c-negation. After doing that, the copula C will be the W -ordinal sum of that copula in the interval mentioned previously. Due to this fact, we can consider f (x) < 1 − x ∀ x ∈ ]0, 1[. In order to construct the absolutely continuous copula C, we make a modification to the copulas Cz,t given by (2). We define the auxiliary function h(x) := 1 − f (x). Given a point z ∈]0, 1[, we construct the sequence {hn (z)}n∈Z , where ⎧  (−1) n+1 ⎪ h (z) , n < 0, h ⎪ ⎨ n h (z) = z, n = 0, ⎪ ⎪ ⎩  n−1  h h (z) , n > 0. Since h(z) < z for z ∈ ]0, 1[ and h is increasing, then we have four possibilities: 1. For n, m ∈ Z such that n < m, we have hn (z) < hm (z), lim hn (z) = 0 and lim hn (z) = 1. n→+∞

n→∞

2. There exists m0 such that hm0 (z) = 0, and lim hn (z) = 1, and for n < m < m0 we have hn (z) < hm (z). n→−∞

3. There exists n0 such that hn0 (z) = 1 and lim hn (z) = 0, and for n0 ≤ n < m we have hn (z) < hm (z). n→∞

4. There exists m0 and n0 such that hm0 (z) = 0 and hn0 (z) = 1, and for n0 ≤ n < m ≤ m0 we have hn (z) < hm (z). In case 1. we have a division of the interval ]0, 1[ in an infinite number of closed intervals. In case 2. the division is in the interval [0, 1[, and the number of intervals is finite to the right, but since hm0 (z) = 0 in the right side, it will be finite in the sense that there is an interval that has 0 as left extreme. Case 3. is the symmetric case of 2. For case 4. there is a division of [0, 1] in a finite number of intervals. We define the set ⎧ Z, in case 1. ⎪ ⎪ ⎨ ] − ∞, m0 − 1] ∩ Z, in case 2. L= in case 3. Z ∩ [n0 , +∞[, ⎪ ⎪ ⎩ [n0 , m0 − 1] ∩ Z, in case 4. The union of the squares of vertices Sn := {(hn (z), 1 − hn (z)), (hn+1 (z), f (hn (z))), (hn (z), 1 − hn (z))), (hn+1 (z), 1 − hn (z))}, for n ∈ L, are a cover of the set {(x, 1 − x) : x ∈ ]0, 1[} (an example of case 1. can be found in Fig. 2). An interior point (x0 , y0 ) of one of these squares S fulfills y0 > f (x0 ). This cover satisfies that the interiors of these squares S are two by two disjoint, so that we can apply the rectangular patchwork construction with the product copula  in each square. If {an }n∈N is a dense sequence in ]0, 1[, we have that the copula constructed, which we denote by D, is given by

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D(x, y) :=

 1 Da (x, y) 2n n

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(3)

n∈N

and it satisfies the properties mentioned previously. This completes the proof. 2 Since the copula D given in the Proof of Proposition 6 is symmetric with respect to the opposite diagonal, we have that its support does not coincide with the closure (which we denote by O for any set O) of the complement of the zero-set (except the trivial case). In the next result we provide conditions on a copula C for which its support coincide with the complement of Z(C), but we need a preliminarytechnical lemma, in which E(f ) will denote the endograph of a function f , i.e. E(f ) := (x, y) ∈ [0, 1]2 : y ≤ f (x) . Lemma 7. Given a c-negation f , there exists a continuous and strictly decreasing c-negation gε such that f (x) ≤ gε (x) and the (Euclidean) distance from the point (x, gε (x)) to E(f ) is less than ε for every point x ∈ [0, 1].  Proof. The function f can be decomposed into the sum of two functions f1 and f2 such that f1 = K n=1 gn , where K can be a natural number or +∞ (the case f2 = 0 corresponds with the case in which the function f is continuous) and gn is the function given by  βn , x ≤ αn gn (x) = 0, αn < x, where 0 < βn < 1 and f2 is a continuous and decreasing function (this is a consequence of the Lebesgue decomposition theorem: see e.g. [29]). We will construct the function gε by following several steps. Step 1. We modify the functions gn in order to obtain functions g n,εn which coincide   with  gn in [0, αn ] ∪ n) n) n) αn + εn (1−α , 1 and by linear interpolation at the points (αn , βn ) and αn + εn (1−α , 0 in αn , αn + εn (1−α . 2n+1 2n+1 2n+1 The election of ε will be done in the order that it gives us the numbering of g . Previously, we impose the condition n n  n∈I εn < ε/2. Fix ε1 such that

g1,ε1 (x) +

K 

gn (x) + f2 (x) < 1 − x.

n=2

In the same way, if the values εn , with n = 1, . . . , k − 1, have been chosen, we choose εn such that k  n=1

gn,εn (x) +

K 

gn (x) + f2 (x) < 1 − x.

n=k+1

If K is finite, the process will end; otherwise, we will continue indefinitely. The sum of the functions gn,εn provides a continuous function f1∗ which satisfies f1∗ (x) ≥ f1 (x). By defining lε = f1∗ + f2 , it follows f (x) ≤ lε (x) and the distance from (x, lε (x)) to E(f ) is less than ε/2 for all x ∈ [0, 1], and lε (x) ≤ 1 − x. If this last inequality is strict for all x, we take gε = lε and the process ends. Step 2. If the process in Step 1. does not end, we take the function lε∗ as lε but replacing the values εn by εn /2. In this case, the described conditions will be given and lε∗ (x) < 1 − x. In any of both cases, the obtained function is denoted by l1,ε . If l1,ε is strictly decreasing, then gε = l1,ε , and the process ends. (−1) Step 3. If there are intervals in which the function is constant, we will consider the quasi-inverse function l1,ε , which is left-continuous. The new function is related to l1,ε so that the intervals in which the function was constant (−1) (−1) have been transformed into discontinuity points of l1,ε . Moreover, l1,ε has no intervals in which it is constant since (−1)

l1,ε is continuous. Now, by applying Steps 1. and 2. to l1,ε we obtain a function that is an homeomorphism which we −1 denote l2,ε . Finally, our function is given by gε = l2,ε . This completes the proof. 2 We now provide the result mentioned above.

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Theorem 8. For a c-negation f , there exists a copula C such that f = ηC and whose support is the closure of the complement of Z(C) if, and only if, f (x) < 1 − x for every x in ]0, 1[. Proof. We prove the result in the case in which the c-negation f is a homeomorphism. Note that the copula D given by (3) is an absolutely continuous copula with a positive density at points (x, y) ∈ [0, 1]2 such that f (x) < y < 1 − (1 − f )−1 (x) and equals zero in the complementary. For the sake of simplicity in notation, we will study the copula B1 (x, y) := x − D(x, 1 − y), which is absolutely continuous and has a positive density at points (x, y) such that h−1 (x) < y < h(x) = 1 − f (x) and zero  in the complementary. The problem is equivalent to constructing an absolutely continuous copula with support (x, y) ∈ [0, 1]2 : y ≤ h(x) . For that, we will use the symmetry with −1 respect to the function e.g. ∈ [0, 1]2 , its symmetric point with respect to  2 h (see  [36]). Given the point (x, h(x))−3 −1 −1 the graph of h is h (x), h (x) , which belongs to the graph of h . We now construct a copula with a positive   density in the set of points (x, y) ∈ [0, 1]2 : h−3 (x) < y < h(x) . For that, let {an }n∈N be a dense sequence in ]0, 1[, and let An,2 be the copula constructed from a rectangular patchwork of B1 in the rectangle of vertices      Rn,2 := (an , h(an )), h2 (an ), h−1 (an ) , an , h−1 (an ) , h2 (an ), h(an )   by using the product copula . Since in the set of points (x, y) ∈ [0, 1]2 : h−1 (x) < y < h(x) the density is positive and it holds h−1 (x) < x ∀ x ∈ ]0, 1[, we have that the margins we apply in the rectangular patchwork are strictly increasing, absolutely continuous and with positive density, and hence the density of An,2 in Rn,2 is positive. Thus, the support of An,2 is (x, y) ∈ [0, 1]2 : h−1 (x) ≤ y ≤ h(x) ∪ Rn,2 . Now we construct the copula B2 (x, y) :=

 1 An,2 (x, y). 2n

n∈N

  The support of this copula is the set (x, y) ∈ [0, 1]2 : h−3 (x) ≤ y ≤ h(x) . Following similar ideas as before, we construct the copula An,3 , but now the symmetry is with respect to the graph of h−3 . Thereby, the function defined by B3 (x, y) :=

 1 An,3 (x, y) 2n

n∈N

  is an absolutely continuous copula whose support is (x, y) ∈ [0, 1]2 : h−7 (x) ≤ y ≤ h(x) and with a positive density   in the set (x, y) ∈ [0, 1]2 : h−7 (x) < y < h(x) . steps as before. As a reIn general, from the copula Bn−1 , we construct the copula Bn using similar  n sult, the support of the copula Bn is (x, y) ∈ [0, 1]2 : h−2 +1 (x) ≤ y ≤ h(x) and it has a positive density in   n (x, y) ∈ [0, 1]2 : h−2 +1 (x) < y < h(x) . Finally, we obtain the absolutely continuous copula L(x, y) :=

 1 Bn (x, y) 2n

(4)

n∈N

  with support (x, y) ∈ [0, 1]2 : y ≤ h(x) . The copula L∗ (x, y) := x − L(x, 1 − y) satisfies the requested conditions. Now we tackle the general case without the restriction that f be continuous. Let gn,ε be functions satisfying conditions in Lemma 7 with ε = 1/n, and we denote them by gn . We construct copulas Ln given by (4) for functions gn . The required copula is K(x, y) :=

 1 Ln (x, y). 2n

n∈N

Since Ln (x, y) = 0 for y ≤ f (x), we have K(x, y) = 0. On the other hand, if (x, y) is a point in [0, 1]2 such that y > f (x), then we have (x, y) ∈ / E(f ). Thus, there exists n0 in N such that the distance from (x, y) to any point of E(f ) is greater than 1/n0 . Therefore, if n > n0 , we have Ln (x, y) > 0 and K(x, y) > 0, which completes the proof. 2

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Fig. 3. From left to right, the supports of the copulas B1 − B4 in the Proof of Theorem 8 for the function f (x) = 1 −

9

√

x.

√ Remark 3. For the function f (x) = 1 − x, the supports of the copulas B1 − B4 in the Proof of Theorem 8 are shown in Fig. 3. We have proved that a given c-negation f , we have, at least, a copula C such that nC = f (recall Theorem 3). Consider the set T = Z(C) ∩ [0, 1]2 \Z(C).

(5)

The set T is non-increasing (we recall that D ⊆ is non-increasing (respectively, non-decreasing) set if for every (x1 , y1 ), (x2 , y2 ) in D with x1 < x2 implies y1 ≥ y2 (respectively, y1 ≤ y2 ). Since there exist copulas C such that C(u, v) = 0 ∀ (u, v) ∈ T , we have that sup{C ∈ C : C(u, v) = 0, ∀(u, v) ∈ T } is a copula (see [48] for details). The mass distribution depends on the shape of T , but we know that it is formed by a singular component concentrated on the graph of T (denoted by Gr(T)) and in a family of non-decreasing sets. In the next example we study a particular case. R2

Example 1. Suppose that f is a convex c-negation (we do not consider the trivial case f (t) = 1 − t ). Let γ = inf{t ∈ a mass in the following manner: a mass γ on ([0, γ ] × [0, 1]) ∩ Gr(f), and a mass [0, 1] : |f  (t)| < 1}. We distribute

 f (γ ) on [0, 1] × 0, f (γ ) ∩ Gr(f). This mass distribution does not provide a copula, since the total mass is less than 1. If μ denotes the measure associated with this mass distribution, then we have D(u, v) := μ[ 0, u] × [0, v]) = 0 if, and only if, (u, v) ∈ E(f ). It is possible to complete the mass in order to obtain a copula. To do that, consider the

marginals of D, which we denote by F and G. Then we have F (u) = u for u ∈ [0, γ ] and G(v) = v for v ∈ 0, f (γ ) . Consequently, the distributions F1 (u) = u − F (u) and G1 (v) = v − G(v) are concentrated on [γ , 1] and f (γ ), 1 , respectively. The distribution D1 (u, v) := M(F1 (u), G1 (v)) is the greatest measure whose marginals are F1 and G1 . The mass associated with D1 is concentrated on [γ , 1] × f (γ ), 1 ⊂ [0, 1]2 \int(E(f )). D is the greatest distribution function with mass concentrated on Gr(f) satisfying that its marginals do not exceed the uniform marginals. Since D(u, v) = D1 (u, v) = 0 ∀ (u, v) ∈ E(f ), we have that Mf (u, v) := D(u, v) + D1 (u, v), (u, v) ∈ [0, 1]2 ,   is the greatest copula such that Z Mf = T . √ As a particular case, consider the c-negation f (t) = 1 − t for all t ∈ [0, 1]. Then we have γ = 1/4,   v, if u ∈ [0, 1/2], u, if u ∈ [0, 1/4], √ G(v) = F (u) = u − 1/4, if u ∈]1/4, 1], 2v − v 2 − 1/4, if u ∈]1/2, 1],   and the greatest copula such that Z Mf = T is given by   √ Mf (u, v) = max 0, min v + u − 1, u − (1 − v)2

(6)

for all (u, v) ∈ [0, 1]2 (see Fig. 4). Copulas based on interpolation on segments connecting the upper boundary curve of the zero-set to the point (1, 1) are called conic copulas [11,24,31,32] (recall also Introduction and Remark 1). On the other hand, we know that, for

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√ Fig. 4. From left to right, the support (in thick lines), the graph and the level curves of the copula Mf for the c-negation f (t) = 1 − t in Example 1.

Fig. 5. Mass distributions of Mf in the cases i. (left) and ii. (right) from Corollary 9.

a given convex c-negation f , the set T given by (5) is the zero-set of the conic copula associated with f . Now we wonder when that conic copula coincides with the copula Mf given by (6). The answer is provided in the following result. In the proof we make use of the concept of nondecreasing set, e.i. a set S in the extended real plane satisfying that if for any (x, y) and (u, v) in S, x < u implies y ≤ v. Corollary 9. The copula Mf given by (6) is a conic copula if, and only if, one of these two cases holds:   a(1−t) i. f (t) = max 1 − (1−a)t with a ∈ [0, 1] and b ∈]0, 1[. b , 1−b ii. f (t) = a(1 − t) with a ∈ [0, 1]. Proof. Let Cf be the conic copula associated with the convex c-negation f . Then the mass of Cf is concentrated in Gr(f) and in the atomic, singular and absolutely continuous cones (see [24]). Since the mass of Mf is concentrated in Gr(f) and in a non-decreasing set given by F1 (u) = G1 (v), we have that Cf has neither absolutely continuous nor singular cone. This implies that f has no absolutely continuous nor singular component. Thus, f must be a polygonal. Therefore, for each point in which f does not exist, the atomic cone has a segment joining that point with point (1, 1). For functions given in i. and ii., the mass distributions are shown in Fig. 5. Note that the polygonal cannot have more than two segments since, in such a case, the segments joining the points in which f does not exist with the point (1, 1) do not constitute a non-decreasing set. 2 3.3. Topological properties of the zero-sets of copulas We say that a sequence of copulas {Cn } converges in the sup-metric to a copula C when d∞ (Cn , C) → 0. In this case we denote Cn → C. Convergence in the sup-metric can be considered for quasi-copulas, as well. Consider the set Z0 := ([0, 1] × {0}) ∪ ({0} × [0, 1]). Let C0 denote the set of copulas C such that Z (C) = Z0 , i.e. the set of copulas without zero-points inside the unit square, and let C1 = C0c , i.e. the complementary of C0 . Our aim is to provide a

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characterization of the relative size (within C) of the set of copulas C such that Z (C) = Z0 . For that, we will use several topological works in the context of the Baire category results (see, e.g., [16,18,19,42]). Let us recall some basic topological definitions. A subset N of a metric space ( , d) is called nowhere dense if its closure has empty interior. A subset A ⊆ is of first category in ( , d) if it can be expressed as (or covered by) a countable union of nowhere dense sets. Finally, A is called residual if Ac = \ A is of first category. Following [7], in complete metric spaces, residual sets are “large sets”, first category sets are “small sets”, and nowhere dense sets are “very small sets”. Theorem 10. The set C1 is not nowhere dense and it is of first category in C. Proof. We will prove that the set C1 is dense in C and, as a consequence, it is not nowhere dense. For C ∈ C, we consider the ordinal sum of W and C in the intervals [0, 1/n] and [1/n, 1], with n ∈ N, and which we denote by Cn . We have that Cn ∈ C1 and Cn → C. Therefore, C1 is dense in C. On the other hand, in order to prove that C1 is of first category in C, we first define the set An = {C ∈ C : C(1/n, 1/n) = 0}. It is immediate that An is a closed set with the distance d∞ . To check that An has no interior points, let C ∈ An and define the family of copulas Ck , with k ∈ N, given by uv + (k − 1)C(u, v) , ∀ (u, v) ∈ [0, 1]2 . k Then we have Ck ∈ C0 and Ck → C. Thus, An is nowhere dense. Furthermore, we have An ⊂ An+1 and ∪n An = C1 , i.e. C1 is of first category in C. 2 Ck (u, v) =

As an immediate consequence of Theorem 10, we have the following result. Corollary 11. The set C0 is residual in C. We have seen that the zero-set of a quasi-copula is also the zero-set of a copula. This fact leads us to the following question: Fixed a zero-set Z, if we define sets CZ = {C ∈ C : Z ⊂ Z(C)} and QZ = {Q ∈ Q : Z ⊂ Z(Q)}, how is the “size” of CZ with respect to QZ ? The answer is provided in the following result. Theorem 12. If Z  Z(W ), then CZ is nowhere dense in QZ . Proof. It is immediate that QZ is a closed set in Q and CZ is a closed set in QZ . We now check that CZ has no interior points in QZ . For that, it suffices to find, for any C ∈ CZ , a sequence {Qn }n∈N of elements in QZ \CZ such that Qn → C. Let C ∈ CZ be fixed. Since Z  Z(W ), there exists a square Su0 ,u1 , with u0 < u1 , and with vertices (u0 , 1 − u0 ), (u1 , 1 − u1 ) , (u1 , 1 − u0 ) and (u0 , 1 − u1 ) satisfying Z ∩ Su0 ,u1 = ∅. We now define a W -ordinal sum of quasi-copulas. In the square Su0 ,u1 we place a replica of a proper quasi-copula E which is not associated with a signed measure (see [25]). We represent this quasi-copula by Wu0 ,u1 ,E . Then we define Wu0 ,u1 ,E (u, v) + (n − 1)C(u, v) , ∀ (u, v) ∈ [0, 1]2 , n which is a proper quasi-copula. Moreover, E ∈ QZ , so that we have Qn ∈ QZ . On the other hand, it is immediate that Qn → C, i.e. C ∈ / int(CZ ), and this completes the proof. 2 Qn (u, v) =

Remark 4. As a consequence of our results, since QZ is a complete metric space endowed with the metric d∞ , we have that CZ is a “very small set” in QZ . Note also that the case Z = Z(W ) is trivial because here we have QZ = CZ = {W }. 3.4. Lattice-theoretic structures of the sets of semi-copulas, quasi-copulas and copulas with a given zero-set We first recall some notions from lattice theory [9]. Given two elementsx and y of a poset (P , ≤), let x ∨ y denote the join (or the least upper bound) of x and y (when it exists); similarly for S, where Sis a subset of P ; x ∧ y denotes the meet (or the greatest lower bound) of x and y (when it exists); and similarly for S. In particular, if S1 and S2

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are two semi-copulas, then (S1 ∨ S2 )(u, v) := max{S1 (u, v), S2 (u, v)} and (S1 ∧ S2 )(u, v) := min{S1 (u, v), S2 (u, v)}. A poset P = ∅ isa lattice if for every x, y in P , x ∨ y and x ∧ y are in P ; and P is a complete lattice if, for every S ⊆ P , S and S exist and are in P . Remark 5. We stress attention upon the fact that the set of c-negations is a lattice by defining pointwise, for every pair of c-negations c1 and c2 , (c1 ∨ c2 )(x) = max {c1 (x), c2 (x)} and (c1 ∧ c2 )(x) = min {c1 (x), c2 (x)}. However, the pointwise supremum of an arbitrary family of c-negations may not be a c-negation. For instance, if for z ∈ [0, 1] we define the function  1 − x, if x ∈ [0, z] , cz (x) = 0, otherwise, then the pointwise supremum of the family {cz : z ∈ [0, α[} is not a c-negation since it is not left-continuous. Despite this, the set of c-negations is a complete lattice. For a negation η, we define the function ηl given by ηl (1) = 0 and ηl (x) = lim η(t). Then it is easy to check that ηl is a c-negation. The supremum (respectively, infimum) of a set of t→x −

c-negations is the c-negation ηl , where η is the pointwise supremum (respectively, infimum). Let S, Q, C and Q\C denote the respective sets of semi-copulas, quasi-copulas, copulas and proper quasi-copulas. The set S is a poset, and it is known that the sets S and Q are complete lattices, but neither C nor Q\C is a lattice (see [13,20,41]). Fixed a zero-set Z of (semi-)copulas, we define the set SZ∗ = {S ∈ S : Z = Z(S)}, and, similarly, we have corresponding sets QZ and Q∗Z , CZ and CZ∗ , and (Q\C)Z and (Q\C)∗Z for quasi-copulas, copulas and proper quasi-copulas, respectively, and SZ = {S ∈ S : Z ⊂ Z(S)}. We now study the lattice-theoretic structure of these sets, where the drastic semi-copula D, given by  0, if (u, v) ∈]0, 1[2 , D(u, v) = M(u, v), otherwise, will be very useful for our purposes. Theorem 13. Given a fixed zero-set Z of semi-copulas, we have the following statements. i. SZ∗ and SZ are lattices. ii. If Z = Z(D), where D is the drastic semi-copula, then SZ∗ is not a complete lattice. iii. SZ is a complete lattice for all Z. Proof. First we prove part i. for the case SZ∗ (the proof for SZ is similar and we omit it). Assume S1 , S2 ∈ SZ∗ . / Z then we have If (u, v) is a point in [0, 1]2 such that (u, v) ∈ Z, then we have (S1 ∧ S2 )(u, v) = 0. If (u, v) ∈ S1 (u, v)S2 (u, v) > 0, which implies (S1 ∧ S2 )(u, v) > 0. Therefore, we have Z(S1 ∧ S2 ) = Z and S1 ∧ S2 ∈ SZ∗ . Using a similar reasoning we obtain S1 ∨ S2 ∈ SZ∗ . To prove part ii., consider S ∈ SZ∗ and the family of semi-copulas Sn given by S(u, v) + (n − 1)D(u, v) , ∀ (u, v) ∈ [0, 1]2 , n  with n ∈ N, whose elements belong to SZ∗ . Then it is easy to check Sn = D, but D ∈ / SZ∗ since Z = Z(D). Finally, we prove part iii.Given a family of semi-copulas Si , with i ∈ I, for an arbitrary and non-empty set of If (u, v) ∈ Z then we have Si (u, v) = 0, whence S(u, v) = 0, i.e. indices I, we have that S = Si is a semi-copula.  Z ⊆ Z(S) and S ∈ SZ . By taking S = Si we also obtain S ∈ SZ , and this completes the proof. 2 Sn (u, v) =

In the next result we will use the rectangular patchwork techniques studied in [21]. Theorem 14. Given a fixed zero-set Z of copulas, we have the following statements: i. If Z = Z(W ), then CZ∗ , CZ , (Q\C)∗Z and (Q\C)Z are not lattices.

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Fig. 6. Supports of the copulas C1 (left) and C2 (right) in the Proof of Theorem 14.

ii. Q∗Z and QZ are lattices. iii. If Z = Z(W ), then Q∗Z is not a complete lattice. iv. QZ is a complete lattice for all Z. Proof. First we prove part i. Let A ∈ CZ∗ , and consider [u1 , u2 ] × [v1 , v2 ] ⊂ [0, 1]2 \Z such that μA ([u1 , u2 ] × [v1 , v2 ]) > 0. Let C1 and C2 be the respective copulas given by C1 (u, v) = min(u, v, max(0, u − 2/3, v − 1/3, u + v − 1)) and C2 (u, v) = C1 (v, u) for all (u, v) ∈ [0, 1]2 (the supports of copulas C1 and C2 are given in Fig. 6). We note that Q = C1 ∨ C2 is a proper quasi-copula (see [41]). Now we apply the rectangular patchwork technique to A in [u1 , u2 ] × [v1 , v2 ] with C1 , and we obtain a copula A1 ∈ CZ∗ . Similarly, we obtain a copula A2 ∈ CZ∗ with C2 . Now, let u , u , v , v be points in [0, 1] such that u1 ≤ u < u , v1 ≤ v < v and  



 μA [u1 , u2 ] × v1 , v μA u1 , u × [v1 , v2 ] 1 = = μA ([u1 , u2 ] × [v1 , v2 ]) μA ([u1 , u2 ] × [v1 , v2 ]) 3 and

 



 μA u1 , u × [v1 , v2 ] μA [u1 , u2 ] × v1 , v 2 = = . μA ([u1 , u2 ] × [v1 , v2 ]) μA ([u1 , u2 ] × [v1 , v2 ]) 3

Note that the existence of such values u , u , v and v is guaranteed by the fact that A is a copula and by the continuity of μA ([u1 , u] × [v1 , v2 ]) and μA ([u1 , u2 ] × [v1 , v]) in u and v, respectively. Then we obtain



 μA ([u1 , u2 ] × [v1 , v2 ]) u , u × v , v = − < 0, 3 i.e. A1 ∨ A2 is a proper quasi-copula, and thus CZ∗ is not a lattice. The proof in the case of CZ is similar, and we omit details. Finally, to prove that (Q\C)∗Z is not a lattice, consider the proper quasi-copulas  (1/2)Q(2u, 2v), (u, v) ∈ [0, 1/2]2 , Q1 (u, v) = M(u, v), elsewhere, μA1 ∨A2



and

 Q2 (u, v) =

(1/2)(1 + Q(2u − 1, 2v − 1)),

(u, v) ∈ [1/2, 1]2 ,

M(u, v),

elsewhere.

With B ∈ CZ∗ , and using similar ideas than above, we have B1, B2 ∈ (Q\C)∗Z , but B1 ∨ B2 ∈ CZ∗ since Q1 ∨ Q2 = M. To prove parts ii., iii. and iv., we use similar arguments to those given in the proof of parts i., ii. and iii. of Theorem 13, so we omit it, and this completes the proof. 2 3.5. Zero-sets of Archimedean copulas Archimedean copulas are a special type of associative copulas, namely 1-Lipschitz continuous t-norms (see [1]). They became popular since they model the dependence structure between risk factors, and are used in many applications, such as finance, insurance, or reliability (see, e.g. [8,37]) due to their simple forms and nice properties.

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Let ϕ : [0, 1] −→ [0, ∞] be a continuous strictly decreasing function such that ϕ(1) = 0. Consider the function Cϕ given by Cϕ (x, y) = ϕ (−1) (ϕ(x) + ϕ(y)),

(x, y) ∈ [0, 1]2 .

(7)

The function given by (7) is a copula if, and only if, ϕ is convex (see [23,26]). Copulas given by (7) are called Archimedean, and ϕ is called the generator of Cϕ . When ϕ(0) = ∞, the Archimedean copula C is said to be strict, and when ϕ(0) < ∞, it is said to be non-strict. When C is strict, C(x, y) > 0 ∀ (x, y) ∈]0, 1]2 .  and W are Archimedean copulas with respective generators ϕ1 (t) = − ln t (with ϕ1 (0) = ∞) and ϕ2 (t) = 1 − t ∀ t ∈ [0, 1]. Moreover,  is strict. In our next result, we provide a sufficient condition for a set to be the zero-set of an Archimedean copula. For that, we need some preliminary known results. A negation is called strong (which we denote by s) if it is an involution, i.e. s(s(u)) = u ∀ u ∈ [0, 1] (see e.g. [5]). If f is a real-valued convex function defined on [a, b], then it is continuous, with the possible exception of a or b. Moreover, f admits right derivative (denoted by f+ ) in ]a, b[ (note that if f is continuous at the point x = a, then f+ (a) exists and, in this case, the value of the derivative can be finite or f (x) → ∞ when x → a + ). Finally, we also have that f+ is monotonically nondecreasing (see e.g. [51]). For the next three lemmas, we refer the reader to [43], [46] and [6], respectively. For the first of the lemmas, recall that given a metric space (E, d), a sequence of functions {fn }n∈N of the space of real-valued continuous functions in E is said to be equicontinuous iff ∀ > 0, there exists δ > 0 such that for every (x, y) ∈ E with d(x, y) < δ then |fn (x) − fn (y)| < , n ∈ N (see e.g. [43]). bounded sequence of Lemma 15 (Arzelà-Ascoli theorem). Suppose that {fn } is an equicontinuous and   pointwise functions from a compact interval [a, b] to R. Then there exists a subsequence fnk that converges uniformly to a continuous function f . Lemma 16 (Schauder fixed-point theorem). In any normed linear space, each compact and convex subset K satisfies the fixed-point property, i.e. every continuous application T : K −→ K has, at least, a fixed point.  17 (Helly selection theorem). For every sequence {Fn } of distribution functions there exists a subsequence Lemma Fnk and a nondecreasing, right-continuous function F such that lim Fnk (x) = F (x) at continuity points x of F . k→∞

We are now in position to provide the result mentioned previously. Theorem 18. A function s is a convex strong negation if, and only if, the set   As = (x, y) ∈ [0, 1]2 : y ≤ s(x) is the zero-set of a non-strict Archimedean copula. Proof. Let Cϕ be an Archimedean copula. If we have Z(Cϕ ) = {({0} × [0, 1]) ∪ ([0, 1] × {0})}, it is due to the fact that Cϕ is strict. Assume Cϕ is non-strict, and consider ϕ(0) = 1. Then we have Z(Cϕ ) = {(x, y) ∈ [0, 1]2 : y ≤ ϕ −1 (1 − ϕ(x))}. It is easy to check that the function s(x) = ϕ −1 (1 − ϕ(x)) is a convex strong negation. Conversely, we need to consider two cases: 1. Suppose s is a convex strong negation with bounded lateral derivatives (note that s must be continuous). Let S be the set of the functions ϕ : [0, 1] −→ [0, 1] satisfying the following conditions: i. ϕ(0) = 1 and ϕ(1) = 0; ii. ϕ is continuous, decreasing and convex; iii. 1 − (ϕ ◦ s)is convex;      (x) ≤ s (0) and (1 − (ϕ ◦ s)) (x) ≤ s (0) for all x in [0, 1[. iv. ϕ+ + + + Note S = ∅ (e.g. ϕ(x) = 1 − x ∈ S), S is convex and, from Lemma 15, it is compact. Furthermore, if ϕ ∈ S, then the function 1 − (ϕ ◦ s) is in S too: Observe that the function 1 − (ϕ ◦ s) satisfies:

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a. 1 − (ϕ ◦ s)(0) = 1 and 1 − (ϕ ◦ s)(1) = 0; b. 1 − (ϕ ◦ s) is continuous and decreasing (since ϕ and s are decreasing and ϕ ◦ s is increasing); c. 1 − (ϕ ◦ s) is convex (which is condition 1.iii. above); = ϕ(x) for all x in [0, 1]; d. 1 − (1 − ϕ(s(s(x))))  = ϕ(s(s(x))) (0)| for all x ∈ [0, 1[ (since ϕ ∈ S); e. (1 − (ϕ ◦ s)) + (x) ≤ |s+   (x)| ≤ |s (0)| for all x in [0, 1[. f. If g = 1 − (ϕ ◦ s), we have (1 − g ◦ s) + (x) = |ϕ+ +  We define the function sˆ : S −→ S as sˆ (ϕ) := 1 − (ϕ ◦ s). Note that for ϕ1 and ϕ2 in S we have sˆ (ϕ1 ) −  sˆ (ϕ2 )∞ = ϕ1 − ϕ2 ∞ , so that sˆ is continuous. From Lemma 16, we have that sˆ has a fixed point, that is, there   exists ϕ ∈ S such that ϕ(x) = 1 − ϕ(s(x)), i.e. ϕ −1 (1 − ϕ(x)) = s(x). Therefore, Z Cϕ = As . (0) = −∞, and let define s as the convex strong negation obtained by linear interpolation of the 2. Suppose s+ n   points (xn , s (xn )), with xn in 2in : i = 0, . . . , 2n . We have |(sn ) + (0)| < +∞, so that there exists a sequence   of of generators {ϕn } such that the sets Asn = (x, y) ∈ [0, 1]2 : y ≤ sn (x) are the zero-sets   the Archimedean copulas Cϕn . Lemma 17 assures that there exists a subsequence ϕσ (n) such that ϕσ (n) → ϕ at points of continuity of ϕ in [0, 1] (we note that Lemma 17 is applied to the functions 1 − ϕn ). The function ϕ satisfies ϕ(0) = 1, ϕ(1) = 0, it is convex in [0, 1], monotone decreasing and continuous at x = 1, since 0 ≤ ϕ(x) ≤ 1 − x for all x in [0, 1]. Moreover, ϕ is continuous at x = 0; otherwise, we would have lim ϕ(x) = κ < 1. In this case, x→0

let a be in [0, 1] such that ϕ(1 − a) < 1 − κ. The pointwise convergence of ϕσ (n) implies that there exists n0 in N such that for any n > n0 we have ϕσ (n) (1 − a) + ϕσ (n) (ξ ) < 1 for any ξ in ]0, 1]. Then Cϕn (1 − a, ξ ) > 0, i.e. sn (1 − a) < ξ , so that s(1 − a) ≤ lim sn (1 − a) < ξ , which implies s(1 − a) = 0, and this is absurd. Therefore n→∞

ϕσ (n) → ϕ pointwise, and then ϕ is continuous. We now check that the convergence is uniform. Let ε > 0. There exists k > 0 such that if |x − y| ≤ 1k then |ϕ(x) − ϕ(y)| < 3ε . Moreover, there exists n0 ∈ N such that for all n > n0 we have        ϕσ (n) i − ϕ i  < ε ,  k k  3

we have i = 0, . . . , k. For x ∈ ki , i+1 k     i+1 i ϕσ (n) ≤ ϕσ (n) (x) ≤ ϕσ (n) , k k which implies     i +1 ε ε i ϕ − ≤ ϕσ (n) (x) ≤ ϕ + . k 3 k 3 Since



i +1 ϕ k



  i ≤ ϕ (x) ≤ ϕ , k

then we obtain       ϕ(x) − ϕσ (n) (x) ≤ ϕ i + ε − ϕ i + 1 + ε ≤ ε, k 3 k 3 i.e. ϕσ (n) → ϕ uniformly.   We conclude that since sσ (n)→ s, ϕσ (n) → ϕ uniformly and ϕσ (n) (x) = 1 − ϕσ (n) sσ (n) (x) , then we have ϕ(x) = 1 − ϕ(s(x)), whence Z Cϕ = As . This completes the proof.

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4. Conclusions In this paper we have studied conditions on sets in order to be zero-sets of semi-copulas, quasi-copulas or copulas. In particular, our main result states that a closed set A is the zero-set for a given (quasi-) copula if, and only if,

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fA (x) := max{y ∈ [0, 1] : (x, y) ∈ A} is a c-negation. Furthermore, we characterize sets that can be the zero-sets of absolutely continuous copulas and those who have the property that they are the support of a copula and they coincide with the closure of the zero-set. Finally, we establish that every convex strong negation determines the zero-set of, at least, one Archimedean copula. Further research can be aimed at providing necessary and sufficient conditions for a set A that coincides with a level set {(x, y) : C(u, v) = t}, with t in ]0, 1[, of some copula C. Generalizations of our results to higher dimensions are also subject of further work. Acknowledgements The authors thank two anonymous referees and Prof. R. Mesiar for helpful comments, and acknowledge the support by the Ministry of Economy and Competitiveness (Spain) under research project MTM2014-60594-P (partially supported by the European Regional Development Fund, ERDF). 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