Coherence and avoidance of sure loss for standardized functions and semicopulas

We discuss avoidance of sure loss and coherence results for semicopulas and standardized functions, i.e., for grounded, 1-increasing functions with value $1$ at $(1,1,\ldots, 1)$. We characterize the existence of a $k$-increasing $n$-variate function $C$ fulfilling $A\leq C\leq B$ for standardized $n$-variate functions $A,B$ and discuss the method for constructing this function. Our proofs also include procedures for extending functions on some countably infinite mesh to functions on the unit box. We provide a characterization when $A$ respectively $B$ coincides with the pointwise infimum respectively supremum of the set of all $k$-increasing $n$-variate functions $C$ fulfilling $A\leq C\leq B$.


Introduction and motivation
In recent literature on statistical reasoning, imprecise probabilities have become one of the main tools for modeling uncertainty, especially in situations when the use of a precise probability model may be questionable or the exact assessment of probability of events impossible.This is often the case in decision making with vague, incomplete, or even conflicting information [20], and in risk management [4].The general theory of imprecise probability [36,5] offers a variety of different models for dealing with imprecision such as lower and upper previsions, lower and upper probabilities, probability boxes, distortion probabilities, capacities, and several others.
An imprecise model is typically required to satisfy some reasonable consistency conditions.Two such conditions are avoidance of sure loss and coherence that were first introduced for lower and upper probabilities [35] and for lower and upper previsions [37].In the behavioural interpretation, avoidance of sure loss means that gambler's assessments of events should not lead to acceptance of bets that would produce net loss, regardless of the outcome.Coherence, on the other hand, suggests that the imprecise assessments should be derived from precise assessments in the sense that each admissible value for the probability of an event should be realizable with a precise probability measure of events.A major difference between the precise and imprecise setting is that lower and upper probabilities are not additive functions.Instead, they are only monotone with respect to set inclusion, i.e., they are capacities.Capacities as a generalization of additive measures were introduced by Choquet in [7] (see also [14] and note that the original definition given there is less general than the one used nowadays).Together with semicopulas they give rise to the framework of universal integrals [16] providing a common frame for many non-additive integrals [8], including the well-known Choquet integral [7,13] and Sugeno integral [33,34,22].
Avoidance of sure loss and coherence were recently translated to the setting of distribution functions of additive probabilities [28], where imprecision is typically modeled with a probability box, i.e., a set of distribution functions bounded pointwise from above and from below.Employing Sklar's theorem, the two notions can also be considered for copulas, which motivated the introduction of imprecise copulas [27] as boxes of copulas bounded by two quasi-copulas C and C. In this setting, avoidance of sure loss is equivalent to the existence of a copula C satisfying C ď C ď C, where ď denotes the pointwise order, while coherence is equivalent to the two conditions Favourable topological properties of copulas facilitated the application of discretization techniques that eventually led to a characterization of the two notions [23,26], given solely in terms of the bounding functions C and C. The newly developed method, now called the ALGEN method [25], was later extended to distribution functions and applied to give a description of probability boxes in terms of coherent imprecise copulas [24,26].
It is natural to ask whether this new characterization can now be translated back to the theory of lower and upper probabilities and previsions.This is one of the motivations for the present paper.We generalize the ALGEN method to the case where the bounds need not be quasi-copulas.Instead, they are only assumed to be grounded, 1-increasing, and have value 1 at p1, 1, . . ., 1q.We will call such functions standardized functions by analogy with [28], where standardized functions were introduced in the setting of distributions.This will allow for our results to be used in the theory of multivariate probability boxes and thus provide a stepping stone towards applications in lower and upper previsions.In particular, our results can be applied to semicopulas (which include distribution functions of capacities with uniform margins [31]), in which case we can omit one of the technical assumptions and also obtain an additional characterization for coherence.The term semicopula was used for the first time by Bassan and Spizzichino [6] in a statistical context.Semicopulas have been known, in a different context, as conjunctors (monotone extensions of the Boolean conjunction with neutral element 1) [9] or as t-seminorms [32].For (structural) properties of the class of semicopulas see [12,11,10].
Furthermore, we also adapt the method to work for classes of functions other than copulas.In particular, we focus on the classes of k-increasing n-quasi-copulas.With k " 2 this includes the class of supermodular n-quasi-copulas.The role of k-increasing n-quasicopulas (especially for k " n ´1) has been investigated in [2] (see also [3]), while the importance of supermodular, sometimes also called L-superadditive, functions has long been recognized, see e.g.[1,19,21,17,18,15,29,30] and the references therein.
The structure of the paper is as follows.In Section 2 we give the necessary definitions and basic properties that will be used throughout the paper.We state our main results on avoidance of sure loss for standardized functions and semicopulas in Section 3 and give their proofs in Sections 4 to 7. Given two specific functions A ď B we construct a kincreasing function C between them on a dense countably infinite mesh by modifying the lower bound A in Section 4 and by modifying the upper bound B in Section 5. We extend the function C to the full unit cube in Section 6 and collect our findings to conclude the proofs of the main results in Section 7. Section 8 is dedicated to results on coherence.

Notions and basic properties
2.1.On k-boxes, multiplicities, and related properties.Throughout the paper we shall denote the unit interval by I " r0, 1s and we will abbreviate the set t1, 2, . . ., nu by rns, where n is an arbitrary positive integer which will be fixed for the whole paper.We also denote the points p0, 0, . . ., 0q P I n by 0 and p1, 1, . . ., 1q P I n by 1.We will use the terms increasing to mean non-decreasing and decreasing to mean non-increasing.Definition 2.1.Choose k P N such that k P rns.Let x " px 1 , x 2 , . . ., x n q P I n and y " py 1 , y 2 , . . ., y n q P I n be two points.A Cartesian product of n closed intervals, i.e., a set of the form rx, ys " rx 1 , y 1 s ˆrx 2 , y 2 s ˆ¨¨¨ˆrx n , y n s will be called a k-box if ˇˇti P rns | x i ă y i u ˇˇ" k and |ti P rns | x i " y i u| " n ´k.
The vertices of a k-box R " rx, ys will be denoted by ver R " ver rx, ys " tx 1 , y 1 u ˆtx 2 , y 2 u ˆ¨¨¨ˆtx n , y n u.
Putting m " |ti P rns | v i " x i u|, the sign of a vertex v of a k-box R " rx, ys is defined by sign R pvq " p´1q m´pn´kq .
The multiplicity of an arbitrary point u P I n with respect to a k-box R is given by m R puq " Note that, given a k-box R " rx, ys, for each vertex v we have n ´k ď m ď n.In particular, sign R pyq " 1, since m " n ´k in this case, and sign R pxq " p´1q k , since m " n.
We denote by R k pI n q the set of all finite disjoint unions of k-boxes with vertices in I n .This means that a typical element R P R k pI n q is of the form R " Ů s j"1 R j , where tR j u s j"1 is an arbitrary finite family (multi-set) of k-boxes with vertices in I n and Ů denotes the formal disjoint union.We extend the definition of the multiplicity of points from a k-box to a finite disjoint union of k-boxes R " Ů s j"1 R j by putting for each u P I n m R puq " Observe that the disjoint unions here are formal disjoint unions, i.e., a priori the k-boxes R j need not be disjoint, we just treat them as such when calculating the multiplicities.Note that this union can also be interpreted as a union of ten 2-boxes, namely, ABED, CDHG, two copies of DEIH, EF JI, HILK, M N QP , OP SR, and two copies of P QT S.
The multiplicities of the points remain the same in this interpretation.This is always the case, because cutting a box into two has no affect on the multiplicities of its points (the multiplicities cancel where the cut is made).
On the right we have a disjoint union of five 2-boxes, namely, ABM K, CDHF , DHN I, JKM L, and M N P O, two of them are crossing.Note that at some points the multiplicities are added, while at others they cancel.In particular, the point M has multiplicity 3 and the point K has multiplicity ´2, while the points D, H, and N have multiplicity 0 due to cancellation.Furthermore, the points E and G have multiplicity 0 since they are no vertices.All other points have multiplicity ´1 or 1.
For a k-box R " rx, ys we have m R puq P t´1, 0, 1u for each u P I n .Notice that this can also hold for a disjoint union of several k-boxes; however, this is a very special case.In fact, the multiplicity of a point with respect to a finite disjoint union of k-boxes can be any integer, as the following lemma shows.Lemma 2.3.Let x P I n be any point which is not a vertex of I n and fix some integer k P rns.Then for every z P Z there exists a finite disjoint union of k-boxes R P R k pI n q such that m R pxq " z.
The point x is a vertex of both R 1 and R 2 , so m R Since the multiplicities of the vertices are additive, for a k-increasing function A : I n Ñ I we also have for any disjoint union of k-boxes R P R k pI n q V A,k pRq " Note that a function A is 1-increasing if and only if it is increasing in each variable.Definition 2.5.An n-variate function A : I n Ñ I is called (i) standardized if it is grounded, 1-increasing, and satisfies Ap1q " 1, (ii) a semicopula if it is grounded, 1-increasing, and has uniform marginals.
We remark that any nonzero, grounded, 1-increasing function A : I n Ñ R can be standardized by dividing it by Ap1q.
The following lemma, though summarizing a very basic mathematical fact, will facilitate arguments and readability in the later proofs.
Lemma 2.6.Consider two n-variate functions A, B : I n Ñ I with A ď B and fix some integer k P rns.For an arbitrary finite disjoint union of k-boxes R P R k pI n q and an arbitrary y P I n the following holds: Because of Lemma 2.6 we have for all R P R k pI n q L pA,Bq k pRq " Here we adopt the convention that the infimum of an empty set equals 0. We also define the functions γ pA,Bq k We want to show that which in turn can be rewritten into We shall investigate the contribution of each d P D to both sides of the inequality (2.3) by distinguishing the two cases (1) d " x, and (2) d ‰ x. Case " 0, its contribution to the righthand side of (2.3) is also |m R 1 pxq| ¨|m R 2 pxq| ¨pBpxq ´Apxqq, and the inequality holds.
Case 2: d ‰ x.Since |m R 2 pxq| ¨mR 1 pdq `|m R 1 pxq| ¨mR 2 pdq " m R 3 pdq, for the contribution of d to the left-hand side of (2.3) we obtain the following lower bound equaling the contribution of d to the right-hand side of (2.3).Thus, inequality (2.2) is verified.The last term of inequality (2.2) is non-negative by assumption, so it follows that In order to obtain the desired result it suffices to take the infimum over all R 1 P R k pDq with m R 1 pxq ă 0, on the one hand, and the infimum over all R 2 P R k pDq with m R 2 pxq ą 0, on the other hand.□ Proof.First assume that x " px 1 , . . ., x n q P s0, 1s n and that B is continuous.If x " 1, then the set tR P R k pI n q | m R pxq ă 0u is empty, P pA,Bq O,k pxq " 0 by definition, and the first inequality holds.If x ‰ 1, there exists x i ă 1.By permuting the coordinates, we may assume without loss of generality that x 1 ă 1. Choose an ε such that 0 ă ε ď 1 ´x1 and denote x 1 " px 1 , 0, . . ., 0, x k`1 , . . ., x n q and x 2 " px 1 `ε, x 2 , . . ., x k , x k`1 , . . ., x n q.Then R since all other vertices of R 1 have at least one coordinate which equals 0. Sending ε to 0 and using the continuity of B gives the first inequality for the point x.
Next assume that x P s0, 1s n and that A is continuous.Choose an ε such that 0 ă ε ď x 1 and put x 1 " px 1 ´ε, 0, . . ., 0, x k`1 , . . ., x n q and x 2 " px Again, by sending ε to 0 and using the continuity of A we obtain the second inequality for the point x.Now assume that x P I n z s0, 1s n , so at least one coordinate x i of x equals 0, implying that Bpxq ´Apxq " 0. We need to prove that P pA,Bq O,k pxq " P pA,Bq M,k pxq " 0. We will do this without using any continuity of A or B, so the same reasoning will work for both cases.Choose an ε ą 0. If the set tR P R k pI n q | m R pxq ă 0u is empty then P pA,Bq O,k pxq " 0 by definition.If it is non-empty then there exists a k-box R 3 " rx 1 , x 2 s with m R 3 pxq " ´1 such that x 1 " px 1  1 , . . ., x 1 n q, x 2 " px 2 1 , . . ., x 2 n q and x 2 j ´x1 j ď ε for all j P rns.We have Since x i " 0 we have x 2 i ď ε, so y i ď ε for every y " py 1 , . . ., y n q P ver R 3 .This means that Bpyq ď Bp1, . . ., 1, y i , 1, . . ., 1q " y i ď ε since B is a semicopula.It follows that P

Main theorems
In this section we formulate our main results, the proofs of which will be given in Section 7. In order to state them we first introduce the following notion.
Condition S. A function A : I n Ñ I satisfies Condition S if there exists a countable set S Ď I such that for every u P I n and every i P rns the set of discontinuities of the section If a function A : I n Ñ I is 1-increasing, then each section t Þ Ñ Apu 1 , . . ., u i´1 , t, u i`1 , . . ., u n q has countably many discontinuities.Condition S requires that all sections have a common countable set of discontinuities.Examples of functions that do respectively do not satisfy Condition S are depicted in Figure 2.Here is our first main result.Theorem 3.1.Let A, B : I n Ñ I be standardized functions with A ď B and fix some integer k P rns.Suppose that at least one of the functions A and B satisfies Condition S for a set S Ď I. Then the following statements are equivalent: (i) There exists a k-increasing n-variate function C : Note that whenever D Ď I n is a countably infinite mesh then the assertion (ii) of Theorem 3.1 implies that L pA,Bq k pRq ě 0 for all R P R k pDq.In the framework of semicopulas, Theorem 3.1 can be strengthened by omitting Condition S. Theorem 3.2.Consider two n-variate semicopulas A, B : I n Ñ I with A ď B and fix some integer k P rns.Then the following statements are equivalent: (i) There exists a k-increasing n-variate semicopula C : Then it follows that A ď A 1 ď B, that the pair pA 1 , Bq satisfies the condition L pA 1 ,Bq k pRq ě 0 for all R P R k pDq, and that γ   In the following proposition we construct C as a pointwise limit of an increasing sequence of functions A piq obtained from A.
Proof.Since D is countable we can arrange the elements of D into a sequence pd i q iPN .We recursively define a sequence of functions A piq : D Ñ I putting A p0q " A and, for i ě 1, pd j q " 0 for all i ě j.Now, let C be the pointwise limit of the sequence pA piq q iPN .The limit exists since at each d P D the sequence of numbers pA piq pdqq iPN is increasing and bounded.It immediately follows that. (i k,D pd j q " 0 for all j P N; (iii) L pC,Bq k pRq " lim iÑ8 L pA piq ,Bq k pRq ě 0 for all R P R k pDq; where (iii) holds because there are only finitely many points d P D with m R pdq ‰ 0, thus completing the proof.□ 4.2.On the k-increasingess of C. We have so far shown that for two n-variate functions A, B : D Ñ I on a dense countably infinite mesh D Ď I n with L pA,Bq k pRq ě 0 for all R P R k pDq we may obtain another function C : D Ñ I n as the pointwise limit of a sequence of functions pA piq q iPN as given by (4.2).
We shall show that the function C obtained in this way is k-increasing, i.e., fulfills L pC,Cq k pRq ě 0 for all R P R k pDq.Before doing so, let us first look at the consequences a violation of the k-increasingness for C would have.
Without loss of generality we may assume that the k-increasingness is violated for a k-box R ˚P R k pDq, i.e., L pC,Cq k pR ˚q ă 0. Note that the k-box R ˚has exactly 2 k vertices, half of them with positive multiplicities.We shall denote these vertices by x i , i.e., for each i P r2 k´1 s we have 3) The following lemma illustrates that for a subset thereof the values of B and C differ and give rise to the existence of a disjoint union of k-boxes with respect to which the vertex has a negative multiplicity.Then there exists s P r2 k´1 s such that for each i P rss there exist a vertex x i P ver R ˚and a finite disjoint union of k-boxes R i P R k pDq with For all other x i P ver R ˚with m R ˚px i q " 1 it holds that Cpx i q " Bpx i q.
Proof.Let R ˚P R k pDq be a k-box with L pC,Cq k pR ˚q " v ă 0, and denote by x i its vertices with positive multiplicity, i.e., for each i P r2 k´1 s we have x i P ver R ˚and m R ˚px i q " 1 (as in (4.3)).
Since γ pC,Bq k,D pdq " mintP pC,Bq O,k,D pdq, Bpdq ´Cpdqu " 0 for all d P D, this holds in particular also for all x i .Assuming that Bpx i q ´Cpx i q " 0 for all vertices x i with i P r2 k´1 s, leads, on the basis of the assumptions for C (compare also Proposition 4.2 (iii)), to the contradiction We may therefore assume that (after a possible rearrangement of the vertices) there exists s P r2 k´1 s such that O,k,D px i q " 0 ă Bpx i q ´Cpx i q if i P rss, Bpx i q ´Cpx i q " 0 if i P r2 k´1 szrss.(4.6) Since for each i P rss we have Bpx i q ą Cpx i q, the point x i is not a vertex of the unit cube I n , and by Lemma 2.3 there exists an R 1 i P R k pDq with m R 1 i px i q ă 0. Since, for all d P D, the function P pC,Bq O,k,D is given by we can further conclude that, for each i P rss, the infimum for P pC,Bq O,k,D px i q is not taken over the empty set, implying that there exists an while m R ˚px i q " 1 for i P rss is trivially fulfilled.□ Under the assumptions of Lemma 4.3 we obtain R i P R k pDq with i P rss for some s P r2 k´1 s, from which additional finite disjoint unions of k-boxes p R and p R i can be constructed.Taking into account that each corresponding vertex x i P ver R ˚fulfills (4.5), and in particular also Cpx i q ă Bpx i q for all i P rss, we define the following additional disjoint unions of k-boxes putting, for i P rss, For every d P D, the multiplicities with respect to p R and p R i , for i P rss, can be evaluated as (4.9)In particular, for each vertex x i of the k-box R ˚with m R ˚px i q " 1 and i P rss we obtain pR ˚q " v ă 0 such that there are vertices x i with m R ˚px i q " 1 and Cpx i q ă Bpx i q for all i P rss and some s P r2 k´1 s, while for all other x i P ver R ˚with m R ˚px i q " 1 the equality Cpx i q " Bpx i q holds.(i) For each i P rss and each p R i as defined by (4.8) we obtain Proof.We first focus on the upper bound for L pC,Bq k p p R i q for some arbitrary but fixed i P rss.Due to C ď B, and taking into account (2.1), we can express leading to the following equivalent expression of (4.11) We shall investigate the contribution of each d P D to both sides of the above equivalent form of inequality (4.11) by distinguishing the following three cases: (1) d " x i , (2) d " x j , with j P rssztiu and (3) d P Dztx j | j P rssu, i.e., whether or not d is one of the vertices of R ˚with positive multiplicity and different values at B and C. Case 1: Suppose d " x i , i.e., m R ˚px i q " 1 and Bpx i q ą Cpx i q (and P pC,Bq O,k,D px i q " 0).We consider two subcases.First assume that m p R i px i q ě 0. By (4.10), for the contribution of x i to L pC,Bq k p p R i q we obtain the following (in)equalities: m p R i px i q ¨Bpx i q " m ¨mR ˚px i q ¨Bpx i q `ÿ lPrssztiu m l ¨mR l px i q ¨Bpx i q " m ¨Bpx i q `ÿ lPrssztiu m l ¨mR l px i q ¨Bpx i q " mpBpx i q ´Cpx i qq `m ¨mR ˚px i q ¨Cpx i q `ÿ lPrssztiu m l ¨mR l px i q ¨Bpx i q ď mpBpx i q ´Cpx i qq `m ¨mR ˚px i q ¨Cpx i q `ÿ lPrssztiu m l ¨maxtm R l px i q ¨Bpx i q, m R l px i q ¨Cpx i qu.
Secondly, if m p R i px i q ă 0, then for the contribution of x i to L pC,Bq k p p R i q we obtain, taking into account Cpx i q ă Bpx i q, m p R i px i q ¨Cpx i q " m ¨mR ˚px i q ¨Cpx i q `ÿ lPrssztiu m l ¨mR l px i q ¨Cpx i q ď m ¨pBpx i q ´Cpx i qq `m ¨mR ˚px i q ¨Cpx i q `ÿ lPrssztiu m l ¨maxtm R l px i q ¨Bpx i q, m R l px i q ¨Cpx i qu.
Case 2: Consider d " x j with j P rssztiu, i.e., m R ˚px j q " 1, and Cpx j q ă Bpx j q.Assuming first that m p R i px j q ě 0 we obtain for the contribution of x j to L pC,Bq k p p R i q the following series of (in)equalities: m p R i px j q ¨Bpx j q " m ¨mR ˚px j q ¨Bpx j q `ÿ lPrssztiu m l ¨mR l px j q ¨Bpx j q " m ¨Bpx j q `mj ¨mR j px j q loooooomoooooon "´m ¨Bpx j q `ÿ lPrsszti,ju m l ¨mR l px j q ¨Bpx j q " ÿ lPrsszti,ju m l ¨mR l px j q ¨Bpx j q ď ÿ lPrsszti,ju m l ¨maxtm R l px j q ¨Bpx j q, m R l px j q ¨Cpx j qu " m ¨mR ˚px j q ¨Cpx j q ´m ¨Cpx j q `ÿ lPrsszti,ju m l ¨maxtm R l px j q ¨Bpx j q, m R l px j q ¨Cpx j qu.
Using the definition of m, it follows that m p R i px j q ¨Bpx j q ď m ¨mR ˚px j q ¨Cpx j q `mj ¨mR j px j q ¨Cpx j q `ÿ lPrsszti,ju m l ¨maxtm R l px j q ¨Bpx j q, m R l px j q ¨Cpx j qu ď m ¨mR ˚px j q ¨Cpx j q `ÿ lPrssztiu m l ¨maxtm R l px j q ¨Bpx j q, m R l px j q ¨Cpx j qu.
Next assume that m p R i px j q ă 0. Then the contribution of x j to L pC,Bq k p p R i q satisfies m p R i px j q ¨Cpx j q " m ¨mR ˚px j q ¨Cpx j q `ÿ lPrssztiu m l ¨mR l px j q ¨Cpx j q ď m ¨mR ˚px j q ¨Cpx j q `ÿ lPrssztiu m l ¨maxtm R l px j q ¨Bpx j q, m R l px j q ¨Cpx j qu.Otherwise m R ˚pdq P t´1, 0u, i.e., m ¨mR ˚pdq ¨Bpdq ď m ¨mR ˚pdq ¨Cpdq.Therefore, evaluating each term separately using (4.9), we obtain Summarizing all three cases we have shown that (4.11) holds since the contribution of any d P D to its left-hand side is surpassed by its contribution to the right-hand side of the inequality.
We now turn to L pC,Bq k p p Rq and show that the inequality (4.12) holds which, following (2.1), can be equivalently expressed by We again look at the contribution of each d P D to both sides of this inequality and distinguish different cases for d P D.
Case 1: d P tx 1 , . . ., x s u, i.e., m R ˚px j q " 1 and Cpx j q ă Bpx j q for each j P rss.Assuming that m p R px j q ą 0, (4.10) implies that also m p R j px j q " m `m p R px j q ą 0. As a consequence, O,k,D px j q " 0, it follows from Proposition 2.9 that Moreover, by (4.11) and taking into account (4.4) and (4.5), we may argue that m p R j px j q ¨pBpx j q ´Cpx j qq ď L pC,Bq k p p R j q ď m ¨pBpx j q ´Cpx j qq `m ¨LpC,Cq k pR ˚q `ÿ lPrssztju m l ¨LpC,Bq k pR l q ă m ¨pBpx j q ´Cpx j qq `m ¨v `ÿ lPrssztju m l ¨|m R l px l q| ¨|v| s " m ¨pBpx j q ´Cpx j qq `m ¨v `m|v| s ps ´1q " m ¨pBpx j q ´Cpx j qq ´m|v| s .
Since m p R j px j q " m `m p R px j q, we further obtain the contradiction 0 ď m p R px j q loomoon ą0 ¨pBpx j q ´Cpx j qq ă ´m ¨|v| s ă 0, showing that necessarily m p R px j q ď 0 for all j P rss.Therefore, for the contribution of any x j with j P rss to the left-hand side of (4.12) we obtain m p R px j q ¨Cpx j q " ˜m ¨mR ˚px j q `s ÿ l"1 m l ¨mR l px j q ¸¨Cpx j q ď m ¨mR ˚px j q ¨Cpx j q `s ÿ l"1 m l ¨maxtm R l px j q ¨Bpx j q, m R l px j q ¨Cpx j qu.
verifying that inequality (4.12) holds.□ We have now collected all the necessary details for showing that the function C obtained as the pointwise limit of the sequence pA piq q iPN does not only have all the properties shown in Proposition 4.2 but is also k-increasing on D. Proposition 4.5.Let D be a dense countably infinite mesh in I n and fix some integer k P rns.Let A, B : D Ñ I be functions with A ď B and L pA,Bq k pRq ě 0 for all R P R k pDq.Furthermore, assume that Apvq " Bpvq for all vertices v of the unit cube I n .Let C : D Ñ I be the pointwise limit of the sequence pA piq q iPN defined in (4.2).Then C is k-increasing on D, i.e., L Assume that C is not k-increasing, i.e., there exists a k-box R ˚such that L pC,Cq k pR ˚q " v ă 0, where the vertices x i of R ˚fulfill m R ˚px i q " 1 and Cpx i q ă Bpx i q for all i P rss and some s P r2 k´1 s, while Cpx i q " Bpx i q for all other vertices with m R ˚px i q " 1.Then, following Lemma 4.3, there exist finite disjoint unions of k-boxes R i P R k pDq such that for all i P rss m R i px i q ă 0 and L pC,Bq k Combining the k-box R ˚and the corresponding disjoint unions of k-boxes R i , we introduce an additional disjoint union p R P R k pDq by means of (4.7).Proposition 4.4 provides us with an upper bound for L pC,Bq k p p Rq by means of (4.12), implying the contradiction showing that C has to be k-increasing on D. □

Construction of C from above and discussion of its properties
The results of the previous section already prove the existence of a k-increasing n-variate function C on a dense countable mesh D by a construction from below, i.e., starting from the lower bound A. A rather natural question is whether or not the construction of a, possibly different, function C could also be initiated from the upper bound B. This question can be answered to the positive.In this section we briefly sketch the proof steps and the construction, pointing to possibly different arguments needed in the proofs in comparison to the results related to the construction of a function C from the lower bound A.
In the single construction step, function B defined on the mesh D is reduced at an arbitrary but fixed point x P D by means of δ pA,Bq k,D , leading to a smaller function B 1 still satisfying L pA,B 1 q k pRq ě 0 for all R P R k pDq.The proof of the following proposition formalizing this step is in complete analogy to the proof of Proposition 4.1.
Proposition 5.1.Let D be a dense countably infinite mesh in I n and fix some integer k P rns.Let A, B : D Ñ I be functions with A ď B and L pA,Bq k pRq ě 0 for all R P R k pDq.Fix a point x P D and define the function B 1 : D Ñ I by Then we have that A ď B 1 ď B, that the pair pA, B 1 q satisfies the condition L pA,B 1 q k pRq ě 0 for all R P R k pDq, and that δ pA,B 1 q k,D pxq " 0.
Since D is a dense countably infinite mesh in I n containing the points 0 and 1, its elements can be rearranged into a sequence pd i q iPN .As a consequence, a function C can be obtained as the pointwise limit of a sequence of functions B piq : D Ñ I recursively defined by successively applying Proposition 5.1, i.e., putting B p0q " B and, for i ě 1, (compare also Proposition 4.2).The function C constructed in this way has the following properties.When showing that C is k-increasing, assume first, to the contrary, that there is some kbox R ˚where the vertices x i P ver R ˚with negative multiplicities, i.e., fulfilling m R ˚px i q " ´1, are in the focus of our argumentation, in particular those fulfilling in addition Apx i q ă Cpx i q with i P rss and some s P r2 k´1 s.Lemma 5.3.Let D be a dense countably infinite mesh in I n and fix some integer k P rns.Let A, C : D Ñ I be functions with A ď C, L pA,Cq k pRq ě 0 for all R P R k pDq, and δ pA,Cq k,D pdq " 0 for all d P D. Furthermore, assume that Apvq " Cpvq for all vertices v of the unit cube I n and that there exists a k-box R ˚P R k pDq with L pC,Cq k pR ˚q " v ă 0. Then there exists s P r2 k´1 s such that for each i P rss there exist a vertex For all other x i P ver R ˚with m R ˚px i q " ´1 the equality Apx i q " Cpx i q holds.
The proof of Lemma 5.3 follows in analogy to the arguments for Lemma 4.3 and is thus omitted at this place.
For any such vertex x i with i P rss, m R ˚px i q " ´1 and m R i px i q ą 0 the following notations and additional disjoint unions of k-boxes can be defined by putting, for i P rss, In analogy to Proposition 4.4, we are now interested in finding upper bounds for L pA,Cq k p q Rq and L pA,Cq k p q R i q.
Proposition 5.4.Let D be a dense countably infinite mesh in I n and fix some integer k P rns.Let A, C : D Ñ I be functions with A ď C, L pA,Cq k pRq ě 0 for all R P R k pDq, and δ pA,Cq k,D pdq " 0 for all d P D. Furthermore, assume that Apvq " Cpvq for all vertices v of the unit cube I n and that there exists a k-box R ˚P R k pDq with L pC,Cq k pR ˚q " v ă 0 such that there are vertices x i with m R ˚px i q " ´1 and Apx i q ă Cpx i q for all i P rss and some s P r2 k´1 s, while for all other vertices x i P ver R ˚with m R ˚px i q " ´1 we have Apx i q " Cpx i q.
(i) For each i P rss and each q R i as defined by (5.3), the following holds (ii) For q R as defined by (5.2) the following holds Proof.When showing the validity of (5.4), the contribution of each d P D to both sides of the inequality can be considered in analogy to the scenario when constructing C from below, i.e., starting from A (compare also the proof of Proposition 4.4).The cases to be checked are (1) d " x i , (2) d " x j with j P rssztiu and (3) d P Dztx j | j P rssu, i.e., distinguishing whether or not d is one of the vertices of R ˚with negative multiplicity and different values with respect to A and C or not.
In order to show that (5.5) holds, only the contribution of d P tx 1 , . . ., x s u, i.e., for elements x j P D with m R ˚px j q " ´1 and Apx j q ă Cpx j q for all j P rss to both sides of the corresponding inequality needs slightly different arguments compared with the situation when constructing C from A (see also the proof of Proposition 4.4).We briefly discuss these differences: Case 1: d P tx 1 , . . ., x s u, i.e., m R ˚px j q " ´1 and Apx j q ă Cpx j q for all j P rss.If m q R px j q ă 0 then also m q R j px j q " ´m `m q R px j q ă 0 and, as a consequence, we get M,k,D px j q " 0, Proposition 2.9 implies |m q R j px j q| ¨pCpx j q ´Apx j qq ď |m q R j px j q| ¨pP pA,Cq O,k,D px j q `P pA,Cq M,k,D px j qq ď L pA,Cq k p q R j q.
Moreover, by (5.4) and in analogy to the proof of Proposition 4.4 we may argue that |m q R j px j q| ¨pCpx j q ´Apx j qq ď m ¨pCpx j q ´Apx j qq `m ¨LpC,Cq ă m ¨pCpx j q ´Apx j qq `m ¨v `ÿ lPrssztju m l ¨|m R l px l q| ¨|v| s " m ¨pCpx j q ´Apx j qq ´m ¨|v| s .
Since m q R px j q " m q R j px j q `m, we obtain the contradiction 0 ď p´1q ¨m q R px j q loomoon ă0 ¨pCpx j q ´Apx j qq " p´m q R j px j q ´mq looooooooomooooooooon "|m | R j px j q|´m ¨pCpx j q ´Apx j qq ă ´m ¨|v| s ă 0, showing that m q R px j q ě 0 for all l P rss.Therefore, for the contribution of any x j with j P rss to the left-hand side of (5.5) we obtain m q R px j q ¨Cpx j q " ˜m ¨mR ˚px j q `s ÿ l"1 m l ¨mR l px j q ¸¨Cpx j q ď m ¨mR ˚px j q ¨Cpx j q `s ÿ l"1 m l ¨maxtm R l px j q ¨Apx j q, m R l px j q ¨Cpx j qu.
Case 2: d P Dztx 1 , . . ., x s u with m R ˚pdq " ´1 and fulfilling Apdq " Cpdq.This case can be handled in analogy to the situation when constructing C from A (see Proposition 5.4).□ Based on these results it can be shown that the function C obtained as pointwise limit of the sequence pB piq q iPN does not only have all the properties mentioned in Proposition 5.2, but is also k-increasing on D. Proposition 5.5.Let D be a dense countably infinite mesh in I n and fix some integer k P rns.Let A, B : D Ñ I be functions with A ď B and L pA,Bq k pRq ě 0 for all R P R k pDq.Furthermore, assume that Apvq " Bpvq for all vertices v of the unit cube I n .Let C : D Ñ I be the pointwise limit of the sequence pB piq q iPN as given by (5.1).Then C is k-increasing on D, i.e., L pC,Cq k pRq ě 0 for all R P R k pDq.
The proof of Proposition 5.5 can be carried out in analogy to the proof of Proposition 4.5, using some of the results of Lemma 5.3 and Propositions 5.2 and 5.4.

From a dense mesh to the unit cube
Given functions A ď B defined on I n , in both Propositions 4.2 and 5.2 a function C, defined on D, is constructed satisfying Apxq ď Cpxq ď Bpxq for all x P D. Propositions 4.5 and 5.5 show that the function C obtained in this way is k-increasing on D for some k P rns.Thus we can extend C to the entire unit cube I n and show that this extension is still kincreasing and lies between A and B (on the whole I n ).C is k-increasing on I n .Let R " rx, ys Ď I n be a k-box.Denote the vertices of R by v 1 , v 2 , . . ., v r , where r " 2 k .We may assume that m R pv j q " 1 if j P r r 2 s and m R pv j q " ´1 if j P rrszr r 2 s.Let ε ą 0. For every j P rrs there exists d j P r0, v j s X D such that p Cpv j q´2 ε r ă Cpd j q.Using these points d j we construct a k-box p R with vertices in D which approximates R. For each i P rns let J 1 i " tj P rrs | pv j q i " x i u and J 2 i " tj P rrs | pv j q i " y i u, so that J 1 i Y J 2 i " rrs.If x i " y i , we define p x i " maxtpd j q i | j P rrsu P δ i and p y i " p x i , so that p x i " p y i ď x i " y i .If x i ă y i , we define p x i " maxtpd j q i | j P J 1 i u P δ i , choose d i P sx i , y i rXδ i , and define p y i " maxtd i , maxtpd j q i | j P J 2 i uu P δ i , so that p x i ď x i ă p y i ď y i .Finally, we put p x " pp x 1 , p x 2 , . . ., p x n q, p y " pp y 1 , p y 2 , . . ., p y n q, and p R " rp x, p ys, the latter being a k-box with vertices in D. Denote the vertices of p R by p d 1 , p d 2 , . . ., p d r in such a way that for all i P rns and j P rrs we have `p d j ˘i " p x i if and only if pv j q i " x i , i.e., if and only if j P J 1 i .Then m p R `p d j ˘" m R pv j q and d j ď p d j ď v j for all j P rrs.By (6.Proof.The functions A and B are semicopulas and C lies between them, so C is grounded and has uniform marginals on D. Since C is grounded and k-increasing on D for some k ě 2, it is also 1-Lipschitz.A 1-Lipschitz function has a unique continuous extension to the closure of its definition set.The fact that D is dense in I n makes the unique extension C 1 defined for all x P I n .Note that C 1 coincides with the extension p C of C defined in (6.1).However, when proving k-increasingness of p C in the proof of Proposition 6.1, Condition S was not utilized.Thus the same proof can be used here to show that C 1 is k-increasing.
Furthermore, since C is 1-Lipschitz on D, the extension C 1 also coincides with the extension r C from the proof of Proposition 6.1.The proofs that Apxq ď r Cpxq and p Cpxq ď Bpxq for all x P I n did not require Condition S, hence Apxq ď C 1 pxq ď Bpxq.□

Proofs of the main theorems
Now we can combine our findings to prove our main results, i.e., Theorems 3.1 and 3.2, which were stated in Section 3.
Proof of Theorem 3.1.Let A, B : I n Ñ I be standardized functions with A ď B. Suppose that at least one of the functions A and B satisfies Condition S with a set S.
We first show that (i) implies (ii).Let C : I n Ñ I be a k-increasing function with To prove that (ii) implies (i), suppose L pA,Bq k pRq ě 0 for all R P R k pI n q.Let D be a dense countably infinite mesh that contains S n .Then L In the proof of Proposition 4.2 we arranged the elements of D into a sequence pd i q iPN .The order is not important for the proof to work.Note, however, that the obtained function C, constructed from below, satisfies Cpd 1 q " A 1 pd 1 q " Apd 1 q `γpA,Bq k,D pd 1 q " Apd 1 q `mintBpd 1 q ´Apd 1 q, P pA,Bq O,k,D pd 1 qu " mintBpd 1 q, Apd 1 q `P pA,Bq O,k,D pd 1 qu.
The proof of Theorem 3.1 could analogously be done with the use of Propositions 5.2 and 5.5, in which case the obtained function C, constructed from above, would satisfy Cpd 1 q " B 1 pd 1 q " Bpd 1 q ´δpA,Bq k,D pd 1 q " maxtApd 1 q, Bpd 1 q ´P pA,Bq M,k,D pd 1 qu.(7.2) Proof of Theorem 3.2.The proof for k ě 2 is analogous to the proof of Theorem 3.1 (note that the assumption k ě 2 appears in Proposition 6.2).The only differences are that we can choose dense countably infinite mesh D arbitrarily since we do not have Condition S, and that we use Proposition 6.2 instead of Proposition 6.1.If k " 1 then the proof that (i) implies (ii) is the same while the opposite direction is trivial since we can take C " A. □

Coherence results
In the previous sections we were dealing with results related to the avoidance of sure loss.Now we present four consequences concerning coherence.Theorems 8.1 and 8.3 consider the case when the bounds are standardized functions, while Theorems 8.2 and 8.4 deal with the case when the bounds are semicopulas.The coherence for the upper bound is given in Theorems 8.1 and 8.2, while the coherence of the lower bound is considered in Theorems 8.3 and 8.4.(iii) for all x P I n : P pA,Bq O,k pxq " Bpxq ´Apxq.Each of the conditions (i), (ii), and (iii) implies that B is a quasi-copula.
Proof.We prove the equivalence of (i) and (ii) the same way as in Theorem 8.1 with the use of Theorem 3.2 instead of Theorem 3.1.
The implication (iii) ñ (i) is trivial.(i) ñ (iii): Assume that condition (i) holds.We have already shown that condition (ii) holds.Every k-increasing semicopula C is also 2-increasing and thus 1-Lipschitz.By (ii) B is a supremum of 1-Lipschitz functions, so it is 1-Lipschitz and thus continuous.By Lemma 2.10 it holds that P (iii) for all x P I n : P pA,Bq M,k pxq " Bpxq ´Apxq.Each of the conditions (i), (ii), and (iii) implies that A is a quasi-copula.

Concluding remarks
We discuss k-increasing n-variate standardized functions, semicopulas and quasi-copulas.The case of k " n refers to the characterization of n-variate copulas and the problem of relating a true copula to an imprecise copula has been solved in [23].In [1,2,3,21] several aspects of k-increasing n-quasi-copulas have been discussed, in particular for k " 2 covering the case of supermodularity.One of our aims has been to further reduce the conditions imposed on the functions we start with.Thus we work with a larger class of functions but still obtain results on their avoidance of sure loss, i.e., provide a characterization of the existence of a k-increasing n-variate standardized function and a semicopula.Furthermore, we could also show coherence results in this general setting.We expect that our results contribute to a deeper understanding of probability and imprecise probabilities.

Definition 2 . 7 .
Let A, B : I n Ñ I be two functions with A ď B and define the function L pA,Bq k

Lemma 2 . 10 .
Let A, B : I n Ñ I be semicopulas with A ď B and fix some integer k P rns.If B is continuous then for all x P I n P pA,Bq O,k pxq ď Bpxq ´Apxq.If A is continuous then for all x P I n P pA,Bq M,k pxq ď Bpxq ´Apxq.

Figure 2 .
Figure 2. Graph of a function which does not satisfy Condition S (left) and a function which satisfies Condition S with S " t 1 3 , 2 3 u (right).

Proposition 4 . 2 .
Let D be a dense countably infinite mesh in I n and fix some integer k P rns.Let A, B : D Ñ I be functions with A ď B and L pA,Bq k pRq ě 0 for all R P R k pDq.Then there exists a function C : D

(4. 2 )
The definition of A piq and Proposition 4.1, imply A ď A p1q ď A p2q ď ¨¨¨ď A pi´1q ď A piq ď ¨¨¨and A piq ď B. Using Proposition 4.1, we also have γ pA piq ,Bq k,D pd i q " 0 for all i P N and L pA piq ,Bq k pRq ě 0 for all R P R k pDq.It follows that γ pA,Bq k,D pdq ě γ pA p1q ,Bq k,D pdq ě ¨¨¨ě γ pA piq ,Bq k,D pdq ě ¨¨¨for all d P D. Hence, γ pA piq ,Bq k,D

Lemma 4 . 3 .
Let D be a dense countably infinite mesh in I n and fix some integer k P rns.Let B, C : D Ñ I be functions with C ď B, L pC,Bq k pRq ě 0 for all R P R k pDq, and γ pC,Bq k,D pdq " 0 for all d P D. Furthermore, assume that Cpvq " Bpvq for all vertices v of the unit cube I n and that there exists a k-box R ˚P R k pDq with L pC,Cq k pR ˚q " v ă 0. (4.4)

. 10 )
Having disjoint unions of k-boxes p R and p R i for i P rss at hand, we are now interested in identifying upper bounds for L pC,Bq k p p Rq and L pC,Bq k p p R i q.Proposition 4.4.Let D be a dense countably infinite mesh in I n and fix some integer k P rns.Let B, C : D Ñ I be functions with C ď B, L pC,Bq k pRq ě 0 for all R P R k pDq, and γ pC,Bq k,D pdq " 0 for all d P D. Furthermore, assume that Cpvq " Bpvq for all vertices v of the unit cube I n and that there exists a k-box R ˚P R k pDq with L pC,Cq k

Case 3 :
Consider d P Dztx 1 , . . ., x s u.If m R ˚pdq " 1 then d is a vertex of R ˚with positive multiplicity and necessarily fulfilling Bpdq " Cpdq.As a consequence, maxtm p R i pdq ¨Bpdq, m p R i pdq ¨Cpdqu " m p R i pdq ¨Cpdq " m ¨mR ˚pdq ¨Cpdq `ÿ lPrssztiu m l ¨mR l pdq ¨Cpdq " m ¨mR ˚pdq ¨Cpdq `ÿ lPrssztiu m l ¨maxtm R l pdq ¨Bpdq, m R l pdq ¨Cpdqu.

Case 2 : 1 m l ¨mR l pdq ¨Bpdq ď m ¨mR ˚pdq ¨Cpdq `s ÿ l" 1 m l ¨mR l pd ¨Bpdqq ď m ¨mR ˚pdq ¨Cpdq `s ÿ l" 1 mR 1 m l ¨mR l pdq ¨Cpdq ď m ¨mR ˚pdq ¨Cpdq `s ÿ l" 1
d P Dztx 1 , . . ., x s u.If m R ˚pdq " 1 then d is a vertex of R ˚with positive multiplicity and necessarily fulfills Bpdq " Cpdq (compare also (4.6)).Then m p R pdq¨Bpdq " m p R pdq ¨Cpdq and m R l pdq ¨Bpdq " m R l pdq ¨Cpdq for all l P rss, and therefore trivially maxtm p R pdq ¨Bpdq, m p R pdq ¨Cpdqu " m p R pdq ¨Cpdq " m ¨mR ˚pdq ¨Cpdq `s ÿ l"1 m l ¨mR l pdq ¨Cpdq " m ¨mR ˚pdq ¨Cpdq `s ÿ l"1 m l ¨maxtm R l pdq ¨Bpdq, m R l pdq ¨Cpdqu.If m R ˚pdq ‰ 1 then d is not a vertex of R ˚or it is a vertex with negative multiplicity to R ˚, i.e., fulfills m R ˚pdq P t0, ´1u since R ˚is a k-box.Assume first that m p R pdq ě 0.Then, due to m R ˚pdq P t0, ´1u, the contribution to the left-hand side of (4.12) satisfies m p R pdq ¨Bpdq " m ¨mR ˚pdq ¨Bpdq `s ÿ l"l ¨maxtm R l pdq ¨Bpdq, m R l pdq ¨Cpdqu.If m p R pdq ă 0 then for the contribution to the left-hand side of (4.12) we obtain m p pdq ¨Cpdq " m ¨mR ˚pdq ¨Cpdq `s ÿ l" pC,Cq k pRq ě 0 for all R P R k pDq.Proof.Note that C " lim iÑ8 A piq implies that for each d P D γ pC,Bq k,D pdq " mintP pC,Bq O,k,D pdq, Bpdq ´Cpdqu " 0, and L pC,Bq k pRq ě 0 for all R P R k pDq, according to Proposition 4.2.

Proposition 5 . 2 .
Let D be a dense countably infinite mesh in I n and fix some integer k P rns.Let A, B : D Ñ I be functions with A ď B and L pA,Bq k pRq ě 0 for all R P R k pDq.Then there exists C : D Ñ I such that (i) A ď C ď B on D, (ii) δ pA,Cq k,D pdq " 0 for all d P D, (iii) L pA,Cq k pRq ě 0 for all R P R k pDq.

Proposition 6 . 1 .
Let A, B : I n Ñ I be standardized functions with A ď B and fix some integer k P rns.Suppose that at least one of the functions A and B satisfies Condition S with set S. Furthermore, let D Ď I n be a dense countably infinite mesh with S n Ď D and C : D Ñ I a k-increasing function such that Apdq ď Cpdq ď Bpdq for all d P D. Then C can be extended to a k-increasing function p C : I n Ñ I such that Apxq ď p Cpxq ď Bpxq for all x P I n .Proof.Suppose first that the function A satisfies Condition S, let D " δ 1 ˆδ2 ˆ¨¨¨ˆδ n and define the function p C : I n Ñ I by p Cpxq " suptCpdq | d P r0, xs X Du. (6.1)Note that the set on the right-hand side is non-empty because of 0 P D. Since C is grounded and k-increasing on D, it is 1-increasing on D. Hence, for each d P D we have p Cpdq " Cpdq, and p C is an extension of C to I n .We claim that p

Proposition 6 . 2 .
Consider two n-variate semicopulas A, B : I n Ñ I with A ď B and fix some integer k P rnszt1u.Further, let D Ď I n be a dense countably infinite mesh and C : D Ñ I a k-increasing function such that Apdq ď Cpdq ď Bpdq for all d P D. Then C can be extended to a k-increasing function C 1 : I n Ñ I such that Apxq ď C 1 pxq ď Bpxq for all x P I n .
pA,Bq k pRq ě 0 for all R P R k pDq.By Proposition 4.2 there exists a function C : D Ñ I defined by (4.2) such that Apdq ď Cpdq ď Bpdq for all d P D. Proposition 4.5 implies that C is k-increasing on D. Hence, by Proposition 6.1, function C can be extended to a k-increasing function C : I n Ñ I such that Apxq ď Cpxq ď Bpxq for all x P I n .□

Theorem 8 . 1 .
Let A, B : I n Ñ I be standardized functions with A ď B and fix some integer k P rns.Suppose that at least one of the functions A and B satisfies Condition S and that L pA,Bq k pRq ě 0 for all R P R k pI n q.Then the following are equivalent:(i) for all x P I n : P pA,Bq O,k pxq ě Bpxq ´Apxq; (ii) for all x P I n :Bpxq " suptCpxq | C : I n Ñ I, A ď C ď B, C is a k-increasing standardized functionu.Proof.(i) ñ (ii): Assume that condition (i) holds and fix some x P I n .Choose a dense countable mesh D Ă I n such that x P D and S n Ď D. Note that for all d P D we haveP pA,Bq O,k pdq " inf RPR k pI n q,We repeat the proof from Theorem 3.1 by choosing d 1 " x in the proof of Proposition 4.2.By Equation (7.1) we get Cpxq " Bpxq since P pA,Bq O,k,D pxq ě P pA,Bq O,k pxq by the above and P pA,Bq O,k pxq ě Bpxq ´Apxq by assumption.Doing this for all x P I n gives us condition (ii), because any k-increasing function between the standardized functions A and B is automatically standardized.(ii) ñ (i): Now assume that condition (ii) holds.Fix some x P I n and some ε ą 0. By condition (ii) there is a k-increasing standardized function C : I n Ñ I such that A ď C ď B and Bpxq ´ε ă Cpxq.Then P pA,Bq O,k pxq ě P pA,Cq O,k pxq and, for any R P R k pI n q with m R pxq ă 0, we have L pxq " inf RPR k pI n q m R pxqă0 L pA,Cq k pRq |m R pxq| ě ´pApxq ´Cpxqq ą Bpxq ´Apxq ´ε and, therefore, P pA,Bq O,k pxq ě Bpxq ´Apxq.□ When the bounds are semicopulas, Condition S can be omitted and an additional equivalent assertion can be given.Theorem 8.2.Let A, B : I n Ñ I be semicopulas with A ď B and fix some integer k P rnszt1u.Assume that L pA,Bq k pRq ě 0 for all R P R k pI n q.Then the following are equivalent: (i) for all x P I n : P pA,Bq O,k pxq ě Bpxq ´Apxq; (ii) for all x P I n : Bpxq " suptCpxq | C : I n Ñ I, A ď C ď B, C is a k-increasing semicopulau; pxq ď Bpxq ´Apxq and (iii) follows.□ The proofs of the following two theorems rely on the proofs of Theorems 3.1 and 3.2 by constructing C from above, replacing the function γ pA,Bq k,D by δ pA,Bq k,D and using Equation (7.2) instead of Equation (7.1) (see also Section 5).Theorem 8.3.Let A, B : I n Ñ I be standardized functions with A ď B and fix some integer k P rns.Suppose that at least one of the functions A and B satisfies Condition S and that L pA,Bq k pRq ě 0 for all R P R k pI n q.Then the following are equivalent: (i) for all x P I n : P pA,Bq M,k pxq ě Bpxq ´Apxq; (ii) for all x P I n : Apxq " inftCpxq | C : I n Ñ I, A ď C ď B, C is a k-increasing standardized functionu.Theorem 8.4.Let A, B : I n Ñ I be semicopulas with A ď B and fix some integer k P rnszt1u.Assume that L pA,Bq k pRq ě 0 for all R P R k pI n q.Then the following are equivalent: (i) for all x P I n : P pA,Bq M,k pxq ě Bpxq ´Apxq; (ii) for all x P I n : Apxq " inftCpxq | C : I n Ñ I, A ď C ď B, C is a k-increasing semicopulau; Two examples of a disjoint union of 2-boxes in I 3 (see Example 2.2).
Example 2.2.Figure1depicts two examples of disjoint unions of 2-boxes in I 3 .On the left we have a disjoint union of four 2-boxes, namely, ABLK, CF JG, M N T S, and OQT R.So this union is comprised of two overlapping pairs of boxes that are disconnected from each other.The multiplicity of the point T is 2 since it is a vertex of two boxes with corresponding multiplicity 1.The points D, E, I, H, and P have multiplicity 0 because they are not vertices of any of the boxes.All other points have multiplicity ´1 or 1.
1pxq P t´1, 1u and, since they differ only in the first coordinate, m R 2 pxq " ´mR 1 pxq.If z ¨mR 1 pxq ą 0 put R " Let k P N be such that k P rns, and let A : I n Ñ I be an n-variate function.Then (i) A is grounded if Apxq " 0 whenever x i " 0 for some i P rns;(ii) A has uniform marginals if Ap1, . . ., 1, x i , 1, . . ., 1q " x i for all x i P I and all i P rns;

as follows. Let R k pDq be the set of all finite disjoint unions of k-boxes with vertices in D and let A, B : D Ñ I be functions with A ď B. Then the functions P
and δ Let δ 1 , δ 2 , . . ., δ n be subsets of I containing both 0 and 1, and put D " ś n i"1 δ i Ď I n .If each set δ i is countably infinite and dense in I then also D is countably infinite and dense in I n .We call such a D a dense countably infinite mesh in I n .Bpdq ´Apdqu.If the point x from Lemma 2.3 belongs to D, then the disjoint union of k-boxes R can be chosen from R k pDq.Furthermore, Lemma 2.6 is valid also if the functions A, B are defined on D only and y P D, in which case we have for all R P R k pDq Let D be a dense countably infinite mesh in I n and fix some integer k P rns.Let A, B : D Ñ I be functions with A ď B and L Bpxq ´Apxq.Proof.If x is a vertex of the unit cube I n then the claim holds, since the right-hand side of the inequality equals 0. So fix x P D which is not a vertex of the unit cube I n .By Lemma 2.3 there exist some R 1 , R 2 P R k pDq with m R 1 pxq ă 0 and m R 2 pxq ą 0 which can be used to define a new disjoint union of k-boxes R 3 by 1), this impliesApx 1 , d 2 , ..., d m , x m`1 , ..., x n q " sup ␣ Apt 1 , d 2 , ..., d m , x m`1 , ..., x n q ˇˇt 1 P r0, x 1 s X δ 1 ( .The functionf 2 : t 2 Þ Ñ Apx 1 , t2 , d 3 , ..., d m , x m`1 , ..., x n q is continuous at t 2 " x 2 , increasing, and δ 2 is dense in I, so Apx 1 , x 2 , d 3 , ..., dContinuing inductively up to index m, and at the last step using the increasing functionf m : t m Þ Ñ Apx 1 , ..., x m´1 , t m , x m`1 , ..., x n q which is continuous at t m " x m ,we obtainApxq " sup ␣ Apx 1 , ..., x m´1 , t m , x m`1 , ..., x n q ˇˇt m P r0, x m s X δ m ( Apt 1 , ..., t m , x m`1 , ..., x n q ˇˇt j P r0, x j s X δ j for all j P rms (completingthe proof when A satisfies Condition S. Suppose now that the function B satisfies Condition S. In this case we define for every x P I n r Cpxq " inftCpdq | d P rx, 1s X Du.The function r C is another extension of C to I n .Similarly as in the previous case we show that also r C is k-increasing.It is obvious that Apxq ď r Cpxq for any x P I n .In order to prove r Cpxq ď Bpxq we use Condition S for the function B to show that Bpxq " inftBpdq | d P rx, 1s X Du in a similar way as above.□ When formulating the counterpart of Proposition 6.1 for the case of semicopulas, we can drop Condition S.