1
Chapter 1
Introduction
This chapter introduces general concepts and relevant results for the present perspective on
aggregation functions based on copulas. In particular, we discuss supermodular functions on
a lattice and we explore some of their basic properties. Various common functional transfor-
mations maintain or generate supermodularity and, above all, there is an equivalence between
supermodularity and a standard notion of complementarity, known also as “increasing dif-
ferences”. The concept of complementarity is well established in economics at least since
Edgeworth and the basic idea of complementarity is that the marginal value of an action is
increasing in the level of other actions avalaible. The mathematical concept of supermodular-
ity formalizes the idea of complementarity and in the literature it is defined for functions on a
general lattice, but our aim is to define supermodularity for aggregation functions.
1.1 Partially Ordered Sets and Lattices
This section introduces and develops concepts and properties involving order and lattices,
by giving also characterizations of sublattice structure.
A lattice is a system of elements with two basic operations: formation of meet and formation
of join, which are respectively denoted by a^ b and a_ b; this notation has been favoured by
Birkhoff and MacLane.
To introduce lattices we define first the relation of partial order and then partially ordered
sets, including chains. Whenever discussing a general partially ordered set, the associated
ordering relation is denoted . Sometimes, the same symbol may be used to denote different
ordering relations on different partially ordered sets, where the particular context precludes any
ambiguities. Any subset of n
is taken to have as the associated ordering relation.
Let L be a set of elements; then a relation of partial order over L is any dyadic relation
over L which is:
(i) reflexive: for every a2 L, a a;
(ii) anti-symmetric: if a b and b a, then a= b;
(iii) transitive: if a b and b c, then a c.
2 Introduction
If a b or b a, we say that a;b are comparable, otherwise a and b are incomparable, in
notation akb.
A set L over which a relation of partial order is defined is called a partially ordered set or
poset.
Notice that elements of a partially ordered set need not be comparable with one another, though
each must be comparable with itself.
If for every pair of elements a;b of a partially ordered set L we have either a b or b a or
both, the set L is said to be simply or totally ordered and is called a chain. Since a b and
b a imply a= b, hence we may define a chain as a partially ordered set in which for every
pair of distinct elements a;b we have either a b or b a (linearity or ordering property). We
note that any subset of a chain is itself a chain. A finite chain of n elements has a least and a
greatest member, and is isomorphic with the sequence of natural numbers(1;2;3;:::;n).
Suppose that L is a partially ordered set and A is a subset of L. If x
0
is in L and x x
0
(x
0
x)
for each x2 A, then x
0
is an upper (lower) bound for A. If x
0
in A is an upper (lower) bound
for A, then x
0
is the greatest (least) element of A. If x
0
is in A and there does not exist any x
00
in A with x
0
x
00
(x
00
x
0
), then x
0
is a maximal (minimal) element of A. A greatest (least)
element is a maximal (minimal) element. A partially ordered set can have at most one greatest
(least) element, but it may have any number of maximal (minimal) elements. Distinct maximal
(minimal) elements do not have ordering property, that is they are unordered. If the set of up-
per (lower) bounds of A has a least (greatest) element, then this least upper bound (greatest
lower bound) of A is the supremum (infimum) of A and is denoted sup
L
(A)(inf
L
(A)) if the
set L is not clear from context or sup(A) (inf(A)) if the set L is clear from context.
If two elements, x
0
and x
00
, of a partially ordered set L have a least upper bound (greatest lower
bound) in L, it is their join (meet) and is denoted x
0
_ x
00
(x
0
^ x
00
). A partially ordered set that
contains the join and the meet of each pair of its elements is a latticehL;_;^i. It shall be
convenient to lay down the convention
W
/ 0=
V
/ 0=?.
A function f(x) from a partially ordered set L to a partially ordered set Y is increasing (de-
creasing) if x
0
x
00
in L implies f(x
0
) f(x
00
)( f(x
00
) f(x
0
)) in Y . A function is monotone if
it is either increasing or decreasing. A function f(x) from a partially ordered set L to a partially
ordered set Y is strictly increasing (strictly decreasing) if x
0
x
00
in L implies f(x
0
) f(x
00
)
( f(x
00
) f(x
0
)) in Y . It is common in the lattice theory literature [10, 97], to use the terms
isotone and antitone rather than “increasing” and “decreasing”, respectively, but the latter are
used herein in order to maintain a more familiar terminology.
1.1.1 Sublattice Structure
If A is a subset of a lattice L and A contains the join and meet (with respect to L) of each
pair of elements of A, then A is a sublattice of L. For a lattice L, letL(L) denote the set of
all nonempty sublattices of L. If A is a sublattice of a lattice L, then A is itself a lattice and in
A the join and meet of any two elements are the same as the join and meet of those same two
elements in L. If L is a lattice, A is a sublattice of L, and A
0
is a sublattice of A, then A
0
is a
sublattice of L. If A is a sublattice of L, L and A are lattices with the same ordering relation,
1.1 Partially Ordered Sets and Lattices 3
i.e. x A
y for x;y2 A implies x L
y, A is not necessarily a sublattice of L.
If f(x) is a function from a set L into a partially ordered set Y , then the level sets of f(x)
on L are the setsfx : x2 L;y f(x)g for y in Y . A function f(x) from a set L into a partially
ordered set Y is a generalized indicator function for a subset A of L if
f(x)=
(
y
00
for x2 A;
y
0
for x2 L and x = 2 A
where y
0
y
00
in Y ; that is, if the only level sets of f(x) on L are L, A and perhaps the empty
set. An indicator function is a generalized indicator function with Y = 1
, y
0
= 0 and y
00
= 1.
If L is a partially ordered set, A is a subset of L and L\[x;¥) is a subset of A for each
x2 A, then A is an increasing set. Equivalently, a subset A of a partially ordered set L is an
increasing set if the indicator function of A\[x;¥) is an increasing function on L for each
x2 A. Increasing sets are useful in characterizing properties of parameterized collections of
distribution functions.
It can be shown that a nonempty finite lattice has a greatest element and a least element
(Lemma 2.2.1 in [98]). A different proof of this result comes from combining the properties
that any nonempty finite partially ordered set has a maximal element and a minimal element
and that a maximal (minimal) element of a lattice is the greatest (least) element.
If L
i
is a partially ordered set with binary relation i
for each i2 I, then the direct product of
these partially ordered sets is the partially ordered set consisting of the set i2I
L
i
with the pro-
duct relation where x
0
x
00
in i2I
L
i
if x
0
i
i
x
00
i
for each i in I. A special case of this direct
product example is the partially ordered set n
=f(x
1
;:::;x
n
) : x
i
2 1
8 i= 1;:::;ng with the
ordering relation where x
0
x
00
in n
if x
0
i
x
00
i
in 1
for i= 1;:::;n. In fact I=f1;:::;ng,
L
i
= 1
with 1
having the usual ordering relation for each i in I and n
= i2I
L
i
.
If L and T are sets and S is a subset of L T , then the section of S at t in T is S
t
=fx : x2
L;(x;t)2 Sg and the projection of S on T isP
T
S=ft : t2 T;S
t
is nonemptyg. Lemma 2.2.2
and Lemma 2.2.3 in [98] show that intersections, sections and projections of sublattices are
also sublattices. Furthermore, the direct product of lattices is a lattice and of sublattices is a
sublattice. Essential properties characterizing sublattices of the direct product of any finite col-
lection of lattices can be expressed in terms of sublattices of the direct product of two lattices.
A lattice in which each nonempty subset A has a supremum_A and an infimum^A is com-
plete. The concept is self-dual and obviously half of the hypothesis is redundant. For a com-
plete lattice L, the supremum of L is denoted by 1 and the infimum of L is denoted by 0. Thus
L is a bounded lattice, with 1 as its greatest element and 0 as its least element.
By Lemma 2.2.1 in [98], any finite lattice is complete. A nonempty complete lattice has a
greatest element and a least element. If A is a sublattice of a lattice L and if, for each nonempty
subset A
0
of A, sup
L
(A
0
) and inf
L
(A
0
) exist and are contained in A, then A is a subcomplete
sublattice of L. By Lemma 2.2.1, any finite sublattice of a lattice is subcomplete. Hence any
sublattice of a finite lattice is subcomplete. Each closed interval in a complete lattice L is
a subcomplete sublattice of L and the supremum and infimum with respect to the closed in-
terval of any subset of the closed interval are the same as the supremum and infimum with
respect to L of that same subset. If L is a lattice and A is a subcomplete sublattice of L, then
4 Introduction
sup
L
(A
0
)= sup
A
(A
0
) and inf
L
(A
0
)= inf
A
(A
0
) for each nonempty subset A
0
of A, A itself is a
complete lattice, and A has a greatest element and a least element if A is nonempty.
1.1.2 Lattice Homomorphisms
A monotone map g : L! M between lattices need not, in general, preserve meets and joins,
but the following statements are equivalent:
g is increasing;
g(x^ y) (g(x)^ g(y))),8x;y2 L;
g(x_ y) (g(x)_ g(y))),8x;y2 L.
The mapping g is called a meet-morphism if g(x^ y)=(g(x)^ g(y))),8x;y2 L and a join-
morphism if g(x_ y)=(g(x)_ g(y))),8x;y2 L. Obviously, meet- and join-morphisms are
increasing mappings, but the converse does not hold. Similarly, the mapping g is called an
inf-morphism if for any non-empty family(x
i
)
i2I
in L it holds that
g(inf
i2I
x
i
)= inf
i2I
g(x
i
)
and a sup-morphism if for any non-empty family(x
i
ji2 I) in L it holds that
g(sup
i2I
x
i
)= sup
i2I
g(x
i
):
If we consider the particular case of a functionj :[0;1])[0;1], the following statements are
equivalent:
j is increasing;
(j(min(x;y))= min(j(x);j(y))),8(x;y)2[0;1]
2
;
(j(max(x;y))= max(j(x);j(y))),8(x;y)2[0;1]
2
.
Clearly, an increasing functionj :[0;1]![0;1] is right-continuous if and only if it is an inf-
morphism, and left-continuous if and only if it is a sup-morphism.
A function g : L! M that preserves finite meets and joins, that is, for which
g(x^ y)=(g(x)^ g(y)))
g(x_ y)=(g(x)_ g(y)))
is called a lattice homomorphism.
1. A lattice monomorphism or lattice embedding is an injective lattice homomorphism;
2. a lattice epimorphism is a surjective lattice homomorphism;
1.2 Increasing differences and supermodular functions 5
3. a lattice endomorphism of L is a lattice homomorphism from L to itself;
4. a lattice isomorphism is a bijective lattice homomorphism, or equivalently, an order iso-
morphism.
Definition 1.1.1 LetL=(L;^
L
;_
L
;0
L
;1
L
) andM=(M;^
M
;_
M
;0
M
;1
M
) be bounded lattices.
A function g : L! M for which g(0
L
)= 0
M
and g(1
L
)= 1
M
is called af0;1g-homomorphism
if g(0
L
)= 0
M
and g(1
L
)= 1
M
.
If L and M are complete lattices, a continuous homomorphism from L to M is a function
h : L! M such that :
h(^A)=^h(A) h(_A)=_h(A)
for every nonempty subset A of L.
1.2 Increasing differences and supermodular functions
Suppose that L and T are partially ordered sets and f(x;t) is a real-valued function on
a subset S of L T . For t in T , let S
t
denote the section of S at t. If f(x;t
00
) f(x;t
0
)
is increasing, decreasing, strictly increasing, or strictly decreasing in x on S
t
00\ S
t
0 for all
t
0
t
00
in T , then f(x;t) has, respectively, increasing differences, decreasing differences,
strictly increasing differences or strictly decreasing differences in (x;t) on S. The con-
ditions of these definitions do not distinguish between the first and second variables because
f(x
0
;t
00
) f(x
0
;t
0
) f(x
00
;t
00
) f(x
00
;t
0
) if and only if f(x
00
;t
0
) f(x
0
;t
0
) f(x
00
;t
00
) f(x
0
;t
00
),
and similarly for a strict inequality.
Suppose that L
i
is a partially ordered set for each i in a set A, L is a subset of i2A
L
i
, an ele-
ment x in L is expressed as x=(x
i
)
i2I
, where x
i
is in L
i
for each i in I and f(x) is a real-valued
function on L. If, for all distinct i
0
and i
00
in I and for all x
0
i
in L
i
for all i in Infi
0
;i
00
g, f(x)
has increasing differences, decreasing differences, strictly increasing differences, or strictly
decreasing differences in (x
i
0;x
i
00) on the section of L atfx
0
i
: i2 Anfi
0
;i
00
gg, then f(x) has,
respectively, increasing differences, decreasing differences, strictly increasing differences,
or strictly decreasing differences on L. If f(x) is differentiable on n
, then f(x) has increas-
ing differences on n
if and only if¶ f(x)=¶x
i
0 is increasing in x
i
00 for all distinct i
0
and i
00
and
all x. If f(x) is twice differentiable on n
, then f(x) has increasing differences on n
if and
only if¶
2
f(x)=¶x
i
0¶x
i
00 0 for all distinct i
0
and i
00
and all x.
Suppose that f(x) is a real-valued function on a lattice L. If
f(x
0
)+ f(x
00
) f(x
0
_ x
00
)+ f(x
0
^ x
00
)
for all x
0
and x
00
in L, then f(x) is supermodular on L. If
f(x
0
)+ f(x
00
)< f(x
0
_ x
00
)+ f(x
0
^ x
00
)
for all unordered x
0
and x
00
in L, then f(x) is strictly supermodular on L. If f(x) is (strictly)
supermodular, then f(x) is (strictly) submodular. A function that is both supermodular and
6 Introduction
submodular is a valuation.
Theorem 2.6.1 and Corollary 2.6.1 in [98] show that a function has increasing differences on
the direct product of a finite collection of chains if and only if the function is supermodular
on that direct product, thereby characterizing supermodularity on the direct product of a finite
collection of chains (and, in particular, on n
) in terms of the nonnegativity of all pairs of
cross–differences. As in Lemma 2.2.4 in [98] the number 2 plays a fundamental role in char-
acterizing sublattice structure, likewise it has a fundamental role for supermodular functions,
since supermodularity on any finite product of chains is equivalent to supermodularity on the
product of each pair of the chains. Thus supermodularity is a second–order property in the
sense that for twice–differentiable functions on n
each class of functions can be character-
ized by certain conditions on the matrix of second partial derivatives.
Theorem 2.6.1 in [98] shows that supermodularity implies increasing differences for a function
on a sublattice of the direct product of lattices. Corollary 2.6.1 in [98] states that increasing dif-
ferences imply supermodularity on the direct product of a finite collection of chains. This result
is a consequence of Theorem 2.6.2 because any real–valued function on a chain is (strictly) su-
permodular. The result of Corollary 2.6.1, characterizing supermodularity in terms of increas-
ing differences, is limited to domains that are the direct product of finitely many chains. Note
that this corollary states that a function is supermodular on the direct product of finitely many
chains if the function is supermodular on each 4-element sublatticefx
0
;x
00
;x
0
_x
00
;x
0
^x
00
g such
that x
0
and x
00
each differ from x
0
^ x
00
in exactly one component.
A function f : 2
! has increasing differences if, for any t t
0
, g(x)= f(x;t) f(x;t
0
) is
an increasing function of x. A function f : S
! has increasing differences if for any s;t
and x, the function
ˆ
f : 2
! ,
ˆ
f( ˆ x
s
; ˆ x
t
)= f(x
s;t
; ˆ x
s
; ˆ x
t
);
obtained by allowing only x
s
and x
t
to vary from x, has increasing differences.
There is an equivalence of supermodularity and increasing differences for a function f : L! ,
where L is a finite dimensional product set L= i2I
L
i
, where each L
i
is a chain in its order i
and L ordered by the product order (see [98], Corollary 2.6.1).
Say that x and y are disjoint, written x? y, if the infimum of x and y is zero, i.e., if x^ y= 0
(see [89], Chapter 5, Section 1). So a mapping f : S
! is supermodular if and only if
it displays increasing differences and so the following proposition characterizes supermodular
functions.
Proposition 1.2.1 A function f : L n
! is supermodular if and only if
f(x+ h+ k) f(x+ k) f(x+ h) f(x) (1.1)
when x;y2 L, for all h;k with h;k 0 , h? k such that x+ h;x+ k;x+ h+ k2 L.
Increasing difference transfers the supermodularity condition to one involving the linear struc-
ture of n
.
It is worth noting that the supermodularity condition is only an “inter-attribute" relation. Intu-
1.2 Increasing differences and supermodular functions 7
itively increasing differences say that there must be “complementarity" between attributes.
Proposition 1.2.2 An n-ary function f : [0;1]
n
![0;1] is supermodular if and only if each of
its two-dimensional sections is supermodular, i.e., for each x2[0;1]
n
and all i; j2f1;2;:::;ng
with i6= j, the function f
x;i; j
: [0;1]
2
![0;1] given by f
x;i; j
(u;v)= f(y), where y
i
= u, y
j
= v
and y
k
= x
k
for k2f1;2;:::;ngnfi; jg, is supermodular.
Proof : On a generic lattice we have the similar result in the Definition 1:2 in [11]. The same
result can be applied to our case as well, because[0;1]
n
is a lattice, 2
Remark 1.2.3 (Supermodular lattices) It is clear that with a supermodular indicator func-
tion we can define a supermodular sublattice in a lattice. This definition and in particular the
description of supermodular sublattices in products of relatively complemented lattices can be
found in [63].
Definition 1.2.4 A sublattice L
0
of a lattice L is called supermodular (in L) if, for any x2 L
0
and y2 L, at least one of the elements x^ y;x_ y belongs to L
0
.
Prototypical examples of supermodular sublattices in any lattice are its ideals, filters, and (set-
theoretic) unions of an ideal and a filter.
In the next chapters we will apply all these concepts in the theory of aggregation functions
and copulas, but, first of all, we are going to introduce the basic background, definitions and
properties in this context.
9
Chapter 2
Aggregation functions and Copulas
Supermodular functions are extensively investigated in different research areas, both pure
and applied. The supermodular property also goes by a variety of names such as L-superadditive
(where L is mnemonic for lattice), superadditive and quasimonotone. Our aim is to apply this
concept to aggregation functions, but, first of all, we recall the basic definitions and properties
both for aggregation functions and copulas. Moreover, we will focus our attention to the main
problem that we have when we want to deal with multivariate copulas as aggregation functions.
2.1 Aggregation Functions
Aggregation operators (also referred to as means or mean operators) correspond to particu-
lar mathematical functions used for information fusion, the broad area that studies methods to
combine data or information supplied by multiple sources. Generally, we consider mathemati-
cal functions that combine a finite number of inputs, called arguments, into a single output.
So, aggregation has for purpose the simultaneous use of different pieces of information pro-
vided by several sources, in order to come to a conclusion or a decision. They are applied in
many different domains and in particular aggregation functions play important role in different
approaches to decision making, where values to be aggregated are typically preference or sa-
tisfaction degrees and thus belong to the unit interval[0;1]. For more details, see [42].
A basic consideration for any aggregation operator is the type of data it is going to fuse. At
present, there exists a large number of aggregation operators applicable to a broad range of data
representation formalisms. For example, aggregation operators on the following formalisms
have been considered in the literature: numerical data, ordinal scales, fuzzy sets, belief func-
tions, among others. The construction of new functions on the basis of new properties or
when considering new knowledge representation formalisms has been studied for a long time.
For example, in the framework of aggregation of preferences, Llull (thirteenth century) and
Nicholas Cusanus (fifteenth century) proposed methods that were later rediscovered by Con-
dorcet and Borda (eighteenth century). They are the Condorcet rule (with the Copeland method
for solving ties) and the Borda count.
Currently, in these early years of the 21st century, an important amount of literature is already
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