# Choquet-Sugeno-like operator based on relation
and conditional aggregation operators

Michał Boczek11footnotemark: 1<EMAIL_ADDRESS>Ondrej
Hutník22footnotemark: 2<EMAIL_ADDRESS>Marek Kaluszka33footnotemark: 3
<EMAIL_ADDRESS>Institute of Mathematics, Lodz University of
Technology, 90-924 Lodz, Poland  Institute of Mathematics, Pavol Jozef Šafárik
University in Košice, 040-01 Košice, Slovakia

###### Abstract

We introduce a Choquet-Sugeno-like operator generalizing many operators for
bounded functions and monotone measures from the literature, e.g., Sugeno-like
operator, Lovász and Owen measure extensions,
$\displaystyle\mathrm{F}$-decomposition integral with respect to a partition
decomposition system, and others. The new operator is based on the concepts of
dependence relation and conditional aggregation operators, but it does not
depend on $\displaystyle t$-level sets. We also provide conditions for which
the Choquet-Sugeno-like operator coincides with some Choquet-like integrals
defined on finite spaces and appeared recently in the literature, e.g. reverse
Choquet integral, $\displaystyle d$-Choquet integral,
$\displaystyle\mathrm{F}$-based discrete Choquet-like integral, some version
of $\displaystyle C_{\mathrm{F}_{1}\mathrm{F}_{2}}$-integral,
$\displaystyle\mathrm{C}\mathrm{C}$-integrals (or Choquet-like Copula-based
integral) and discrete inclusion-exclusion integral. Some basic properties of
the Choquet-Sugeno-like operator are studied.

###### keywords:

Choquet integral; Sugeno integral; Conditional aggregation operator; Monotone
measure; Decomposition integral; Möbius transform

##  1 Introduction

The origin of investigation of nonadditive integrals goes back to the works of
Vitali, Choquet, Shilkret and Sugeno. Their works have been extensively
studied and several generalizations have been proposed in recent years
grouping the mentioned integrals into two main classes: depending on
$\displaystyle t$-level set and independent on it. The first group includes
universal integral [23], upper and lower $\displaystyle n$-Sugeno integral
[3], whereas the second group includes copula-based integrals [23, 26],
$\displaystyle\mathrm{F}$-decomposition integrals [20] and inclusion-exclusion
integral [17]. Obviously there are some functionals belonging to both groups
such as seminormed fuzzy integrals [36] and the prominent nonadditive
integrals of Choquet [11] and Sugeno [37]. Especially, generalizations of the
discrete Choquet integral have recently attracted the greatest interest. The
discrete Choquet integral can be equivalently expressed as follows

$\displaystyle\displaystyle(C)\int\limits_{X}f\,{\mathrm{d}}\mu$
$\displaystyle\displaystyle=\sum_{i=1}^{n}f_{(i)}\cdot(\mu(B_{(i)})-\mu(B_{(i+1)})),$
(1)
$\displaystyle\displaystyle=\sum_{i=1}^{n}(f_{(i)}-f_{(i-1)})\cdot\mu(B_{(i)}),$
(2)
$\displaystyle\displaystyle=\sum_{i=1}^{n}\bigl{(}f_{(i)}\cdot\mu(B_{(i)})-f_{(i-1)}\cdot\mu(B_{(i)})\bigr{)},$
(3) $\displaystyle\displaystyle=\sum_{D\in
2^{X}\setminus\\{\emptyset\\}}\textsf{Mob}_{\mu}(D)\cdot\min_{i\in D}f(i),$
(4)

where $\displaystyle X=\\{1,2,\dots,n\\},$ $\displaystyle\mu$ is a monotone
measure on $\displaystyle 2^{X},$ integrand $\displaystyle f$ is a vector with
nonnegative entries, $\displaystyle(\cdot)\colon X\to X$ is a permutation such
that $\displaystyle 0=f_{(0)}\leqslant f_{(1)}\leqslant\ldots\leqslant
f_{(n)},$ and $\displaystyle B_{(i)}=\\{(i),\ldots,(n)\\}$ for $\displaystyle
i\in\\{1,2,\dots,n\\}$ with $\displaystyle B_{(n+1)}=\emptyset.$ Moreover,
$\displaystyle\textsf{Mob}_{\mu}$ is the Möbius transform of
$\displaystyle\mu.$ Replacing the product operation in its standard form (1),
equivalent form (2), expanded form (3), and the Möbius transform form (4) by
some other fusion functions with appropriate properties one can obtain a
resulting aggregation-like function providing various generalizations of the
discrete Choquet integral. For the present state-of-art of the generalizations
of the discrete Choquet integral we refer to [12].

As far as we know, there is no unified setting for a common generalization of
all the expressions (1)–(4) of the discrete Choquet integral. In recent paper
[8] authors write in Conclusion: “(…) the possible extension of our
generalization idea to Choquet integrals expressed in terms of Möbius
transform does not seem so easily achievable, as it is not clear how the
information provided by a restricted dissimilarity function can be included in
such representation. Nevertheless, it is worth to mention that, in any case,
the equivalence between the different possible representations of the standard
Choquet integral will be most probably lost in our more general setting.”
Therefore, in this paper we try to cover some generalizations of expressions
(1)–(4) of the discrete Choquet integral in one general formula. For this
purpose we introduce a Choquet-Sugeno-like operator (see Definition 3.2)
independent on $\displaystyle t$-level sets, but depending upon a conditional
aggregation operator and some relation between sets in a collection. These are
two novel ingredients in comparison with the known approaches existing in the
literature. Due to these new elements we may provide many well-known and
examined operators as well as several new operators not yet studied in the
literature. A detail discussion is included in Examples 3.4–3.9 and in Section
4, where we also give relationships between the existing generalizations of
discrete Choquet integral and our Choquet-Sugeno-like operator. As a by-
product, we join the works that simultaneously generalize the Sugeno integral
and the Choquet integral on finite sets, similarly to the works [20, 26, 31,
39].

The paper is organized as follows. In the forthcoming section we provide basic
notations and definitions we work with. In Section 3 we give several examples
of Choquet-Sugeno-like operator known in the literature. In the next Section 4
we provide the conditions for which our Choquet-Sugeno-like operators
coincides with the operators extending the formulas (1)–(4). In Section 5 we
examine some basic properties of the Choquet-Sugeno like operator such as
monotonicity, homogeneity, subadditivity, convexity and idempotency.

##  2 Basic notations

Let $\displaystyle(X,\Sigma)$ be a measurable space, where
$\displaystyle\Sigma$ is a $\displaystyle\sigma$-algebra of subsets of
$\displaystyle X.$ In what follows,
$\displaystyle\Sigma_{0}=\Sigma\setminus\\{\emptyset\\}.$ A monotone or
nonadditive measure on $\displaystyle\Sigma$ is a finite nondecreasing set
function $\displaystyle\mu\colon\Sigma\to[0,\infty),$ i.e.,
$\displaystyle\mu(C)\leqslant\mu(D)$ whenever $\displaystyle C\subset D$ with
$\displaystyle\mu(\emptyset)=0,$ where “$\displaystyle\subset$” and
“$\displaystyle\subseteq$” denote the proper inclusion and improper inclusion,
respectively. We denote the class of all monotone measures on
$\displaystyle(X,\Sigma)$ by $\displaystyle\mathbf{M}.$ If
$\displaystyle\mu(X)=1,$ then $\displaystyle\mu$ is called a capacity and
$\displaystyle\mathbf{M}^{1}$ denotes the set of all capacities. For
$\displaystyle\mu,\nu\in\mathbf{M},$ we write $\displaystyle\mu\leqslant\nu$
whenever $\displaystyle\mu(D)\leqslant\nu(D)$ for any $\displaystyle
D\in\Sigma.$ For $\displaystyle X=\\{1,2,\ldots,n\\},$ we say that
$\displaystyle\mu\in\mathbf{M}$ is symmetric if the condition
$\displaystyle|C|=|D|$ implies $\displaystyle\mu(C)=\mu(D),$ where
$\displaystyle|E|$ is the cardinality of a set $\displaystyle E.$
Additionally, a capacity $\displaystyle\mu$ is symmetric whenever
$\displaystyle\mu(\cdot)=\nu(\cdot)/\nu(X)$ with
$\displaystyle\nu\in\mathbf{M}$ being symmetric. Denote by
$\displaystyle\widehat{\mathbf{M}}$ the family of all set functions
$\displaystyle\widehat{\mu}\colon\Sigma\to(-\infty,\infty)$ with
$\displaystyle\widehat{\mu}(\emptyset)=0.$

By $\displaystyle\mathbf{F}$ denote the set of all
$\displaystyle\Sigma$-measurable (measurable, for short) nonnegative bounded
functions on $\displaystyle X$ and
$\displaystyle\mathbf{F}^{1}=\\{f\in\mathbf{F}\colon\textstyle{\sup_{x\in
X}f(x)}\leqslant 1\\}.$ We write $\displaystyle f\leqslant g$ if
$\displaystyle f(x)\leqslant g(x)$ for all $\displaystyle x\in X.$ Increasing
[resp. nondecreasing] function $\displaystyle f\in\mathbf{F}$ means that
$\displaystyle f(x)<f(y)$ [resp. $\displaystyle f(x)\leqslant f(y)$] whenever
$\displaystyle x<y.$ We say that a function $\displaystyle f\in\mathbf{F}$ is
subadditive if $\displaystyle f(x+y)\leqslant f(x)+f(y)$ for any
$\displaystyle x,y$ such that $\displaystyle x+y\in X.$ For $\displaystyle
t\geqslant 0,$ define the $\displaystyle t$-level set of a function
$\displaystyle f\in\mathbf{F}$ as $\displaystyle\\{f\geqslant t\\}=\\{x\in
X\colon f(x)\geqslant t\\}.$

Let $\displaystyle D_{1},D_{2}\subseteq(-\infty,\infty).$ We say that an
operation $\displaystyle\circ\colon D_{1}\times D_{2}\to[0,\infty)$ is
nondecreasing if $\displaystyle a\circ b\leqslant c\circ d$ for any
$\displaystyle(a,b),(c,d)\in(D_{1},D_{2})$ such that $\displaystyle a\leqslant
c$ and $\displaystyle b\leqslant d.$ We say that $\displaystyle x\mapsto
x\circ b$ is subadditive for any $\displaystyle b$ if $\displaystyle(x+y)\circ
b\leqslant x\circ b+y\circ b$ for any $\displaystyle b.$ In a similar way we
define the subadditivity of $\displaystyle x\mapsto a\circ x.$

The set $\displaystyle\\{1,\ldots,k\\}$ is denoted by $\displaystyle[k].$ Let
$\displaystyle\mathbbm{1}_{D}$ denote the indicator function of a set
$\displaystyle D,$ that is, $\displaystyle\mathbbm{1}_{D}(x)=1$ if
$\displaystyle x\in D$ and $\displaystyle\mathbbm{1}_{D}(x)=0$ otherwise, and
$\displaystyle\mathbbm{1}(S)$ denote the indicator function of a logical
sentence $\displaystyle S,$ that is, $\displaystyle\mathbbm{1}(S)=1$ if
$\displaystyle S$ is true and $\displaystyle\mathbbm{1}(S)=0$ otherwise. For
any $\displaystyle a,b\in[0,\infty),$ let $\displaystyle a\wedge
b=\min\\{a,b\\}$ and $\displaystyle a\vee b=\max\\{a,b\\}$ as well as
$\displaystyle(x)_{+}=x\vee 0$ for any $\displaystyle x\in(-\infty,\infty).$
Moreover, $\displaystyle{\mathbb{N}}=\\{1,2,\ldots\\}$ denotes the set of
natural numbers and $\displaystyle 0_{D}(x)=0$ for all $\displaystyle x\in D,$
where $\displaystyle D\in\Sigma.$ We adopt the usual conventions:
$\displaystyle\textstyle{\sum_{k\in\emptyset}f(k)=0}$ and
$\displaystyle\textstyle{\sum_{i=j}^{k}a_{i}=0}$ for $\displaystyle k<j.$

A crucial concept used in this paper is an extension of aggregation functions
introduced recently in [2].

######  Definition 2.1.

A map $\displaystyle\mathsf{A}(\cdot|D)\colon\mathbf{F}\to[0,\infty)$ is said
to be a conditional aggregation operator with respect to $\displaystyle
D\in\Sigma_{0}$ if it satisfies the following conditions:

  1. $\displaystyle(C1)$

$\displaystyle\mathsf{A}(f|D)\leqslant\mathsf{A}(g|D)$ for any $\displaystyle
f,g\in\mathbf{F}$ such that $\displaystyle f(x)\leqslant g(x)$ for all
$\displaystyle x\in D$;

  2. $\displaystyle(C2)$

$\displaystyle\mathsf{A}(\mathbbm{1}_{D^{c}}|D)=0.$

The nonempty set $\displaystyle D$ will be called a conditional set.

From Definition 2.1 it follows that
$\displaystyle\mathsf{A}(f|D)=\mathsf{A}(f\mathbbm{1}_{D}|D)$ for any
$\displaystyle f\in\mathbf{F},$ so the value $\displaystyle\mathsf{A}(f|D)$
can be interpreted as “an aggregated value of $\displaystyle f$ on
$\displaystyle D$”. In other words, the conditional aggregation operator only
depends on the value of the considered function defined on the conditional
set. Conditional aggregation operator extends the concept of aggregation
operator $\displaystyle\mathsf{Agg}(\cdot)$ presented by Calvo et al. [9] to
all measurable functions. For some examples and methods of construction of
conditional aggregation operators we refer to [2]444 Observe that in [2] the
conditional aggregation operator can take infinite values as well. To avoid
unnecessary complications, we consider here only finite-valued conditional
aggregation operators.. By
$\displaystyle{\mathscr{A}}=\\{\mathsf{A}(\cdot|D)\colon D\in\Sigma\\}$ we
denote a family of conditional aggregation operators (FCA in short). In order
for the FCA to be well defined for all the sets from $\displaystyle\Sigma,$
from now on we consider the conditional aggregation operators with the
additional assumption $\displaystyle\mathsf{A}(\cdot|\emptyset)=0.$ Several
important FCAs $\displaystyle{\mathscr{A}}$ will be highlighted using the
superscript such as
$\displaystyle{\mathscr{A}}^{\inf}=\\{\mathsf{A}^{\inf}(\cdot|D)\colon
D\in\Sigma\\}$ and
$\displaystyle{\mathscr{A}}^{\sup}=\\{\mathsf{A}^{\sup}(\cdot|D)\colon
D\in\Sigma\\},$ where
$\displaystyle\mathsf{A}^{\inf}(f|D)=\textstyle{\inf_{x\in D}f(x)}$ and
$\displaystyle\mathsf{A}^{\sup}(f|D)=\textstyle{\sup_{x\in D}f(x)}$ for any
$\displaystyle D\in\Sigma_{0}.$ In order to avoid ambiguity in the markings,
we still assume that
$\displaystyle\mathsf{A}^{\inf}(f|\emptyset)=0=\mathsf{A}^{\sup}(f|\emptyset)$
when necessary.

##  3 Choquet-Sugeno-like operator and its several special cases

In this section we introduce an operator which is based on two families of
conditional aggregation operators and a relation between the conditional sets,
which is the main ingredient providing new possibilities. Conditional sets
will be chosen from a collection being any subset
$\displaystyle{\mathcal{D}}\subseteq\Sigma_{0}.$ A nonempty family
$\displaystyle{\mathcal{H}}$ of collections will be called a decomposition
system, i.e., $\displaystyle{\mathcal{H}}\subseteq
2^{\Sigma_{0}}\setminus\\{\emptyset\\}$ (cf. [20]). Several decomposition
systems often used in the paper are summarized in the following example.

######  Example 3.1.

  1.   2. (a)

$\displaystyle{\mathcal{H}}_{\text{one}}=\\{\Sigma_{0}\\}$ is a singleton
consisting of the maximal collection;

  3. (b)

$\displaystyle{\mathcal{H}}_{\text{part}}=\\{{\mathcal{P}}\colon{\mathcal{P}}\text{
is a~{}finite partition of }X\\}.$ We say that $\displaystyle{\mathcal{P}}$ is
a finite partition of $\displaystyle X$ if
$\displaystyle{\mathcal{P}}=\\{D_{1},\ldots,D_{n}\\}$ such that
$\displaystyle\textstyle{\bigcup_{i=1}^{n}D_{i}=X}$ and $\displaystyle
D_{i}\cap D_{j}=\emptyset$ for any $\displaystyle i\neq j$ and $\displaystyle
D_{i}\in\Sigma_{0}$ for any $\displaystyle i$;

  4. (c)

$\displaystyle{\mathcal{H}}_{\text{chain}}=\\{{\mathcal{D}}_{l}\colon
l\in{\mathbb{N}}\\},$ where $\displaystyle{\mathcal{D}}_{l}$ is a chain of
length $\displaystyle l$ defined as
$\displaystyle{\mathcal{D}}_{l}=\\{D_{l},\ldots,D_{1}\\},$ where
$\displaystyle D_{l}\subset\ldots\subset D_{1}$ for any $\displaystyle
D_{i}\in\Sigma_{0}$ and all $\displaystyle i\in[l].$ For instance, for
$\displaystyle X=[2]$ we have
$\displaystyle{\mathcal{H}}_{\text{chain}}=\big{\\{}\\{\\{1\\}\\},\\{\\{2\\}\\},\\{\\{1,2\\}\\},\\{\\{1\\},\\{1,2\\}\\},\\{\\{2\\},\\{1,2\\}\\}\big{\\}}.$

We say that $\displaystyle{\mathcal{R}}$ is a relation on
$\displaystyle{\mathcal{D}}\cup\\{\emptyset\\}$ if
$\displaystyle{\mathcal{R}}\subseteq({\mathcal{D}}\cup\\{\emptyset\\})\times({\mathcal{D}}\cup\\{\emptyset\\}),$
where $\displaystyle{\mathcal{D}}$ is a collection. For two sets
$\displaystyle C,D\in{\mathcal{D}}\cup\\{\emptyset\\}$ being in relation
$\displaystyle{\mathcal{R}}$ we write $\displaystyle(C,D)\in{\mathcal{R}}.$
Although the relation $\displaystyle{\mathcal{R}}$ depends on a collection
$\displaystyle{\mathcal{D}},$ we will not indicate this dependence explicitly
in the notation.

Now we can define the Choquet-Sugeno-like operator. Note that we do not use
the word “integral”, as there is no unambiguous definition of an integral in
the literature, see [2, 10, 23]. Since this paper is related to another
aspects of this topic, we leave this problem for further discussions.

######  Definition 3.2.

Let $\displaystyle{\mathcal{H}}$ be a decomposition system,
$\displaystyle{\mathcal{D}}$ a collection from $\displaystyle{\mathcal{H}},$
and $\displaystyle{\mathcal{R}}$ a relation on
$\displaystyle{\mathcal{D}}\cup\\{\emptyset\\}.$ Then for
$\displaystyle\mathrm{L}\colon[0,\infty)^{3}\times(-\infty,\infty)\to(-\infty,\infty)$
the Choquet-Sugeno-like operator of $\displaystyle f\in\mathbf{F},$
$\displaystyle\mu\in\mathbf{M}$ and
$\displaystyle\widehat{\mu}\in\widehat{\mathbf{M}}$ is defined by

$\displaystyle\displaystyle\mathrm{CS}^{\mathrm{L}}_{{\mathcal{H}},{\mathscr{A}},\widehat{{\mathscr{A}}}}(f,\mu,\widehat{\mu})=\sup_{{\mathcal{D}}\in{\mathcal{H}}}\sum_{(C,D)\in{\mathcal{R}}}\mathrm{L}\big{(}\mathsf{A}(f|C),\widehat{\mathsf{A}}(f|D),\mu(C),\widehat{\mu}(D)\big{)},$
(5)

where $\displaystyle{\mathscr{A}}$ and $\displaystyle\widehat{{\mathscr{A}}}$
are FCAs.

######  Remark 3.3.

Note that it is sufficient to define $\displaystyle\mu$ and
$\displaystyle\widehat{\mu}$ on
$\displaystyle\textstyle{\bigcup_{{\mathcal{D}}\in{\mathcal{H}}}{\mathcal{D}}}$
instead of $\displaystyle\Sigma.$

Let us underline once again that the relation $\displaystyle{\mathcal{R}}$ in
(5) depends on a collection $\displaystyle{\mathcal{D}}$ although we do not
write it. Now we show that the Choquet-Sugeno-like operator generalizes many
concepts from the literature.

######  Example 3.4 (upper Sugeno-like operator).

Let $\displaystyle\mathrm{F}\colon[0,\infty)^{2}\to[0,\infty),$
$\displaystyle{\mathcal{R}}=\\{(D,D)\colon D\in{\mathcal{D}}\\}$ for
$\displaystyle{\mathcal{D}}\in{\mathcal{H}}=\\{\\{D\\}\colon
D\in\Sigma_{0}\\}$ and $\displaystyle\mathrm{L}(x,y,z,w)=\mathrm{F}(x,z).$
Then (5) takes the form

$\displaystyle\displaystyle\mathrm{CS}^{\mathrm{F}}_{{\mathcal{H}},{\mathscr{A}}}(f,\mu)=\sup_{D\in\Sigma_{0}}\mathrm{F}(\mathsf{A}(f|D),\mu(D)).$
(6)

Putting $\displaystyle\mathsf{A}(\cdot|D)=\mathsf{A}^{\inf}(\cdot|D)$ for any
$\displaystyle E\in\Sigma_{0}$ we obtain the upper Sugeno-like operator

$\displaystyle\displaystyle\mathrm{Su}^{\mathrm{F}}(f,\mu)=\sup_{D\in\Sigma_{0}}\mathrm{F}(\inf_{x\in
D}f(x),\mu(D)).$ (7)

In particular, for $\displaystyle\mathrm{F}\colon[0,1]^{2}\to[0,1]$ being a
fuzzy conjunction555A binary function
$\displaystyle\circ\colon[0,1]^{2}\to[0,1]$ is called a fuzzy conjunction if
it is nondecreasing and fulfils $\displaystyle 0\circ 0=0\circ 1=1\circ 0=0$
and $\displaystyle 1\circ 1=1.$ the operator (7) is the $\displaystyle
q$-integral [13], whereas for $\displaystyle\mathrm{F}\colon[0,1]^{2}\to[0,1]$
being a semicopula666A binary function
$\displaystyle\circ\colon[0,1]^{2}\to[0,1]$ is called a semicopula if it is
nondecreasing and fulfils $\displaystyle 1\circ a=a\circ 1=a$ for any
$\displaystyle a\in[0,1].$ the seminormed fuzzy integral [4, 5, 6, 7, 36] of
$\displaystyle(f,\mu)\in\mathbf{F}^{1}\times\mathbf{M}^{1}$ is recovered. The
upper Sugeno-like operator with $\displaystyle\mathrm{F}=\wedge$ is the famous
Sugeno integral [37], whereas for $\displaystyle\mathrm{F}=\cdot$ we get the
Shilkret integral [35]. If $\displaystyle\mathrm{F}$ is nondecreasing, arguing
as in the proof of [4, Theorem 2.2], the upper Sugeno-like operator (7) can be
rewritten in term of $\displaystyle t$-level sets in the following way

$\displaystyle\displaystyle\mathrm{Su}^{\mathrm{F}}(f,\mu)=\sup_{t\geqslant
0}\mathrm{F}(t,\mu(\\{f\geqslant t\\}))$ (8)

known as the (upper) generalized Sugeno integral [10, 22]. For $\displaystyle
X=[n]$ the operator (8) takes the form

$\displaystyle\displaystyle\mathrm{Su}^{\mathrm{F}}(f,\mu)=\max_{i\in[n]}\mathrm{F}(f_{(i)},\mu(B_{(i)})),$
(9)

where $\displaystyle(\cdot)\colon[n]\to[n]$ is a permutation such that
$\displaystyle f_{(1)}\leqslant\ldots\leqslant f_{(n)}$ and $\displaystyle
B_{(i)}=\\{(i),\ldots,(n)\\}$ for $\displaystyle i\in[n],$ as
$\displaystyle\mathrm{F}$ is nondecreasing. The operator presented in (9) was
studied by Horanská and Šipošová [18] for $\displaystyle\Sigma=2^{[n]},$
$\displaystyle\mathrm{F}\colon[0,1]^{2}\to[0,1]$ and
$\displaystyle(f,\mu)\in\mathbf{F}^{1}\times\mathbf{M}^{1}.$

######  Example 3.5 (generalized Lebesgue integral for sum).

Let $\displaystyle\mathrm{L}(x,y,z,w)=\mathrm{F}(x,z).$ Then (5) with
$\displaystyle{\mathscr{A}}={\mathscr{A}}^{\inf}$ and
$\displaystyle{\mathcal{R}}=\\{(D,D)\colon D\in{\mathcal{D}}\\}$ for
$\displaystyle{\mathcal{D}}\in{\mathcal{H}}$ can be rewritten as follows

$\displaystyle\displaystyle\mathrm{CS}^{\mathrm{F}}_{{\mathcal{H}},{\mathscr{A}}^{\inf}}(f,\mu)=\sup_{{\mathcal{D}}\in{\mathcal{H}}}\sum_{D\in{\mathcal{D}}}\mathrm{F}\big{(}\mathsf{A}^{\inf}(f|D),\mu(D)\big{)}.$
(10)

Putting $\displaystyle\mathrm{F}(x,z)=x\otimes z$ and
$\displaystyle{\mathcal{H}}={\mathcal{H}}_{\text{part}}$ in (10) we get

$\displaystyle\displaystyle\mathrm{CS}^{\otimes}_{{\mathcal{H}}_{\text{part}},{\mathscr{A}}^{\inf}}(f,\mu)=\sup_{{\mathcal{D}}\in{\mathcal{H}}_{\text{part}}}\sum_{D\in{\mathcal{D}}}\mathsf{A}^{\inf}(f|D)\otimes\mu(D)$
(11)

the generalized Lebesgue integral with $\displaystyle\oplus=+$ defined in [42,
Definition 3.1], where $\displaystyle\otimes$ is a pseudo-multiplication (see
[42, Definition 2.2]). Let $\displaystyle\otimes=\cdot$ and $\displaystyle
X=(a,b],$ where $\displaystyle a<b$ and $\displaystyle
a,b\in(-\infty,\infty).$ Consider

$\displaystyle{\mathcal{H}}_{\text{part}}^{\ast}=\\{\\{I_{1},\ldots,I_{k}\\}\colon
k\in{\mathbb{N}}\\}\subset{\mathcal{H}}_{\text{part}},$

where $\displaystyle I_{i}=(x_{i},x_{i+1}]$ with the measure
$\displaystyle\mu_{G}(I_{i})=G(x_{i+1})-G(x_{i})$ for a nondecreasing function
$\displaystyle G$ on $\displaystyle[a,b].$ Then (11) is the lower Darboux-
Stieltjes integral of the form

$\displaystyle\displaystyle\mathrm{CS}^{\cdot}_{{\mathcal{H}}_{\text{part}}^{\ast},{\mathscr{A}}^{\inf}}(f,\mu_{G})=\sup_{{\mathcal{D}}\in{\mathcal{H}}_{\text{part}}^{\ast}}\sum_{I_{i}\in{\mathcal{D}}}\mu_{G}(I_{i})\cdot\inf_{x\in
I_{i}}f(x).$

Setting
$\displaystyle{\mathcal{H}}={\mathcal{H}}_{\text{part}}^{\ast\ast}=\\{\\{X\\}\\}$
and $\displaystyle\mathrm{F}(x,z)=x\cdot z$ in (10) we get the min-max
integral [38, Theorem 4.13]

$\displaystyle\mathrm{CS}^{\cdot}_{{\mathcal{H}}_{\text{part}}^{\ast\ast},{\mathscr{A}}^{\inf}}(f,\mu)=\mu(X)\cdot\inf_{x\in
X}f(x).$

######  Example 3.6 ($\displaystyle\mathrm{F}$-decomposition integral with
respect to $\displaystyle{\mathcal{H}}_{\text{part}}$).

Let $\displaystyle X=[n]$ and $\displaystyle\Sigma=2^{[n]}.$ Observe that the
operator (10) with $\displaystyle{\mathcal{H}}={\mathcal{H}}_{\text{part}}$
and a nondecreasing function $\displaystyle\mathrm{F}$ takes the form

$\displaystyle\displaystyle\mathrm{CS}^{\mathrm{F}}_{{\mathcal{H}}_{\text{part}},{\mathscr{A}}^{\inf}}(f,\mu)=\mathcal{I}^{\mathrm{F}}_{{\mathcal{H}}_{\text{part}},\mu}(f),$
(12)

where

$\displaystyle\displaystyle\mathcal{I}^{\mathrm{F}}_{{\mathcal{H}},\mu}(f)=\sup\Big{\\{}\sum_{D\in{\mathcal{D}}}\mathrm{F}(a_{D},\mu(D))\colon\sum_{D\in{\mathcal{D}}}a_{D}\mathbbm{1}_{D}\leqslant
f\,\,{\mathcal{D}}\in{\mathcal{H}}\Big{\\}}$ (13)

is an $\displaystyle\mathrm{F}$-decomposition integral. Setting
$\displaystyle\mathrm{F}=\cdot$ in (12) we obtain the Pan-integral based on
the standard arithmetic operation addition and multiplication [41]. The
operator (13) is studied in [20] for
$\displaystyle\mathrm{F}\colon[0,1]^{2}\to[0,\infty)$ and
$\displaystyle(f,\mu)\in\mathbf{F}^{1}\times\mathbf{M}^{1}.$

The assumption $\displaystyle{\mathcal{H}}={\mathcal{H}}_{\text{part}}$ in
(12) is essential to prove the equivalence between formula (10) and (12) for
any nondecreasing binary function $\displaystyle\mathrm{F}$ and any
$\displaystyle(f,\mu)\in\mathbf{F}\times\mathbf{M}.$ Indeed, let
$\displaystyle X=[3]$ and
$\displaystyle{\mathcal{H}}=\\{\\{D_{1},D_{2},D_{3}\\}\\}\neq{\mathcal{H}}_{\text{part}},$
where $\displaystyle D_{1}=\\{1\\},$ $\displaystyle D_{2}=\\{1,3\\}$ and
$\displaystyle D_{3}=X.$ Assume that $\displaystyle\mu(C)=1$ for any
$\displaystyle\emptyset\neq C\subseteq X,$ $\displaystyle f(1)=0.4,$
$\displaystyle f(2)=0.2$ and $\displaystyle f(3)=0.3.$ Then (10) with
$\displaystyle\mathrm{F}(x,y)=xy$ takes the form
$\displaystyle\mathrm{CS}^{\cdot}_{{\mathcal{H}},{\mathscr{A}}^{\inf}}(f,\mu)=\textstyle{\sum_{i=1}^{3}\mathsf{A}^{\inf}(f|D_{i})}=0.9.$
On the other hand, (13) has the form

$\displaystyle\displaystyle\mathcal{I}^{\cdot}_{{\mathcal{H}},\mu}(f)$
$\displaystyle\displaystyle=\sup\Big{\\{}\sum_{i=1}^{3}a_{D_{i}}\colon\sum_{i=1}^{3}a_{D_{i}}\mathbbm{1}_{D_{i}}\leqslant
f\Big{\\}}\leqslant f(1),$

since the condition
$\displaystyle\textstyle{\sum_{i=1}^{3}a_{D_{i}}\mathbbm{1}_{D_{i}}\leqslant
f}$ implies that $\displaystyle a_{D_{1}}+a_{D_{2}}+a_{D_{3}}\leqslant f(1),$
and so
$\displaystyle\mathrm{CS}^{\cdot}_{{\mathcal{H}},{\mathscr{A}}^{\inf}}(f,\mu)>\mathcal{I}^{\cdot}_{{\mathcal{H}},\mu}(f).$

######  Example 3.7 (generalization of the Lovász extension).

Let $\displaystyle X=[n]$ and $\displaystyle\Sigma=2^{[n]}.$ Assume that
$\displaystyle{\mathcal{R}}=\\{(D,D)\colon D\in{\mathcal{D}}\\}$ for
$\displaystyle{\mathcal{D}}\in{\mathcal{H}}_{\text{one}},$
$\displaystyle\mathrm{L}(x,y,z,w)=y\circ w$ and
$\displaystyle\widehat{\mu}(D)=\textsf{Mob}_{\mu}(D)$ with
$\displaystyle\mu\in\mathbf{M},$ where
$\displaystyle\textsf{Mob}_{\mu}(D)=\textstyle{\sum_{C\subseteq
D}(-1)^{|D\setminus C|}\mu(C)}$ is the Möbius transform ([15, Section 2.10]).
Then (5) is a generalization of the Lovász extension,

$\displaystyle\displaystyle\mathrm{CS}^{\circ}_{{\mathcal{H}}_{\text{one}},{\mathscr{A}}}(f,\textsf{Mob}_{\mu}{})=\sum_{\emptyset\neq
D\subseteq X}\mathsf{A}(f|D)\circ\textsf{Mob}_{\mu}(D).$ (14)

In particular, for $\displaystyle\circ=\cdot$ and
$\displaystyle\mathsf{A}(\cdot|D)=\mathsf{A}^{\inf}(\cdot|D)$ we get the
discrete Choquet integral expressed in terms of the Möbius transform, known as
the Lovász measure extension [25]. On the other hand, for the product
conditional aggregation operator
$\displaystyle\mathsf{A}^{\textrm{prod}}(f|D)=\textstyle{\prod_{i\in D}f(i)},$
the formula (14) is the generalized Owen extension of $\displaystyle\mu.$ The
original Owen measure extension corresponds to
$\displaystyle\mathsf{A}^{\textrm{prod}}(\cdot|\cdot)$ and
$\displaystyle\circ=\cdot,$ see [33]. Assume that
$\displaystyle\circ\colon[0,1]\times(-\infty,\infty)\to(-\infty,\infty)$ is
bounded on $\displaystyle[0,1]^{2},$
$\displaystyle(f,\mu)\in\mathbf{F}^{1}\times\mathbf{M}^{1}$ and
$\displaystyle\mathsf{A}(h|D)=\mathsf{Agg}(h\mathbbm{1}_{E})$ is an
aggregation operator (see [9]). Then the operator (14) with

  * •

$\displaystyle\circ=\cdot$ is studied by Kolesárová et al. [24],

  * •

$\displaystyle\mathsf{Agg}(h\mathbbm{1}_{E})=\textstyle{\inf_{x\in E}h(x)}$ is
examined by Fernandez et al. [14],

  * •

$\displaystyle\circ$ with the values in $\displaystyle[0,1]$ is discussed by
Horanská [19].

######  Example 3.8.

Let $\displaystyle X=[n]$ and $\displaystyle\Sigma=2^{[n]}.$ Put
$\displaystyle{\mathcal{R}}=\\{(D,D^{c})\colon D\in{\mathcal{D}}\\}$ for
$\displaystyle{\mathcal{D}}\in{\mathcal{H}}_{\text{one}},$
$\displaystyle\mathrm{L}(x,y,z,w)=(x-y)_{+}\circ z,$
$\displaystyle{\mathscr{A}}={\mathscr{A}}^{\inf}$ and
$\displaystyle\widehat{{\mathscr{A}}}={\mathscr{A}}^{\sup}.$ Then (5) takes
the form

$\displaystyle\displaystyle\mathrm{CS}^{\mathrm{L}}_{{\mathcal{H}}_{\text{one}},{\mathscr{A}}^{\inf},{\mathscr{A}}^{\sup}}(f,\mu)=\sum_{\emptyset\neq
D\subseteq X}(\min_{x\in D}f(x)-\max_{x\in D^{c}}f(x))_{+}\circ\mu(D).$ (15)

For $\displaystyle\circ=\cdot\,,$ we get an alternative representation of the
Choquet integral presented in [21], see also [1, p. 149].

######  Example 3.9.

(Generalized $\displaystyle p$-variation) Let
$\displaystyle{\mathcal{R}}\subseteq{\mathcal{D}}\times{\mathcal{D}}$ and
$\displaystyle\mathrm{L}(x,y,z,w)=|x-y|^{p},$ where $\displaystyle p\geqslant
1.$ Then the operator (5) with
$\displaystyle{\mathscr{A}}=\widehat{{\mathscr{A}}},$ called the generalized
$\displaystyle p$-variation, takes the form

$\displaystyle\displaystyle{\mathrm{V}}^{p}_{{\mathscr{A}}}(f)=\sup_{{\mathcal{D}}\in{\mathcal{H}}}\sum_{(C,D)\in{\mathcal{R}}}\big{|}\mathsf{A}(f|C)-\mathsf{A}(f|D)\big{|}^{p}.$

The well known notion of $\displaystyle p$-variation we get when taking
$\displaystyle X=[a,b],$ $\displaystyle\Sigma=2^{X},$
$\displaystyle\mathsf{A}(f|\\{x\\})=f(x)$ and
$\displaystyle{\mathcal{R}}=\\{(\\{x_{i-1}\\},\\{x_{i}\\})\colon\\{x_{i}\\}\in{\mathcal{D}},\,i\in[n]\\},$
where
$\displaystyle{\mathcal{D}}\in{\mathcal{H}}=\\{\\{\\{x_{0}\\},\\{x_{1}\\},\ldots,\\{x_{k}\\}\\}\colon
a=x_{0}<x_{1}<\ldots<x_{n-1}<x_{n}=b,\,n\in{\mathbb{N}}\\}.$

######  Remark 3.10.

General properties of operators defined in (6), (10), (14) and (15) have not
been studied in the literature so far.

##  4 Connections with other operators with respect to monotone measure:
finite space case

In [8, 18, 20, 23, 26, 27, 28, 29, 30] the authors study properties of
operators (1)–(3) mainly by replacing the product by another binary function.
So, it is natural to find a connection between them and the Choquet-Sugeno-
like operator. We will describe them in this section. Additionally, we provide
a relationship between the Choquet-Sugeno-like operator and the discrete
inclusion-exclusion integral [17].

In this section we assume that $\displaystyle X=[n]$ and
$\displaystyle\Sigma=2^{[n]}$ with $\displaystyle n\geqslant 2.$ Moreover, to
shorten the notation, we introduce two relations
$\displaystyle{\mathcal{R}}^{+}$ and $\displaystyle{\mathcal{R}}^{-}$ on
$\displaystyle{\mathcal{D}}_{l}\cup\\{\emptyset\\}$ (see Example 3.1 (c)) as
follows:

  * •

$\displaystyle{\mathcal{R}}^{+}=\big{\\{}(D_{1},D_{2}),(D_{2},D_{3}),\ldots,(D_{l},D_{l+1})\big{\\}}$;

  * •

$\displaystyle{\mathcal{R}}^{-}=\big{\\{}(D_{1},D_{0}),(D_{2},D_{1}),\ldots,(D_{l},D_{l-1})\big{\\}},$

where $\displaystyle D_{l}\subset D_{l-1}\subset\ldots\subset D_{1}$ and
$\displaystyle D_{0}=D_{l+1}=\emptyset.$

### Connection with $\displaystyle{\mathfrak{C}}_{\circ}^{\mu,\widehat{\mu}}$

Let $\displaystyle\circ\colon[0,\infty)^{2}\to[0,\infty).$ The operator
$\displaystyle{\mathfrak{C}}_{\circ}^{\mu,\widehat{\mu}}$ is defined as
follows

$\displaystyle\displaystyle{\mathfrak{C}}_{\circ}^{\mu,\widehat{\mu}}(f)=\sum_{i=1}^{n}f_{(i)}\circ(\mu(B_{(i)})-\widehat{\mu}(B_{(i+1)}))$
(16)

for any
$\displaystyle(f,\mu,\widehat{\mu})\in\mathbf{F}\times\mathbf{M}\times\mathbf{M}$
such that $\displaystyle\mu\geqslant\widehat{\mu},$ where
$\displaystyle(\cdot)\colon[n]\to[n]$ is a permutation such that
$\displaystyle f_{(1)}\leqslant\ldots\leqslant f_{(n)},$ $\displaystyle
B_{(i)}=\\{(i),\ldots,(n)\\}$ for $\displaystyle i\in[n]$ and $\displaystyle
B_{(n+1)}=\emptyset.$ The operator (16) depends on the permutation, which need
not be unique in general. Therefore, in order the operator
$\displaystyle{\mathfrak{C}}_{\circ}^{\mu,\widehat{\mu}}$ to be well defined,
its evaluation must not depend on the considered permutation. Observe that if
$\displaystyle f$ is either decreasing or increasing, then the permutation is
unique. This means that in the class of all decreasing or increasing
functions, the operator
$\displaystyle{\mathfrak{C}}_{\circ}^{\mu,\widehat{\mu}}$ is well defined for
any binary operation $\displaystyle\circ$ and
$\displaystyle\mu,\widehat{\mu}\in\mathbf{M}.$ If we assume that the map
$\displaystyle x\mapsto a\circ x$ is Lebesgue measurable for any
$\displaystyle a,$ then using the same argument as Horanská and Šipošová [18,
Proposition 12] one can show that the operator (16) is well defined for any
$\displaystyle(f,\mu,\widehat{\mu})\in\mathbf{F}\times\mathbf{M}\times\mathbf{M}$
if and only if $\displaystyle a\circ b=g(a)b$ for some function $\displaystyle
g\colon[0,\infty)\to[0,\infty).$ Also the special case of (16) with
$\displaystyle\circ\colon[0,1]^{2}\to[0,1],$ $\displaystyle
f\in\mathbf{F}^{1}$ and $\displaystyle\mu=\widehat{\mu}\in\mathbf{M}^{1}$ is
examined in [18].

######  Proposition 4.1.

The operator (16) is well defined for any $\displaystyle f\in\mathbf{F}$ and
any binary operation $\displaystyle\circ\colon[0,\infty)^{2}\to[0,\infty)$
whenever $\displaystyle\mu,\widehat{\mu}\in\mathbf{M}$ are symmetric such that
$\displaystyle\mu\geqslant\widehat{\mu}.$

In terms of conditional aggregation operators the operator (16) can be
rewritten as

$\displaystyle\displaystyle{\mathfrak{C}}_{\circ}^{\mu,\widehat{\mu}}(f)=\sum_{i=1}^{n}\mathsf{A}^{\inf}(f|B_{(i)})\circ(\mu(B_{(i)})-\widehat{\mu}(B_{(i+1)})).$
(17)

This form resembles the Choquet-Sugeno-like operator (5) with
$\displaystyle{\mathcal{H}}={\mathcal{H}}_{\text{chain}},$
$\displaystyle{\mathcal{R}}={\mathcal{R}}^{+},$
$\displaystyle{\mathscr{A}}={\mathscr{A}}^{\inf},$
$\displaystyle\mu,\widehat{\mu}\in\mathbf{M}$ such that
$\displaystyle\mu\geqslant\widehat{\mu}$ and
$\displaystyle\mathrm{L}(x,y,z,w)=x\circ(z-w)_{+},$ i.e.,

$\displaystyle\displaystyle\mathrm{CS}^{\mathrm{L}}_{{\mathcal{H}}_{\text{chain}},{\mathscr{A}}^{\inf}}(f,\mu,\widehat{\mu})=\sup_{{\mathcal{D}}\in{\mathcal{H}}_{\text{chain}}}\sum_{i=1}^{l}\mathsf{A}^{\inf}(f|D_{i})\circ(\mu(D_{i})-\widehat{\mu}(D_{i+1}))$
(18)

under the convention that $\displaystyle D_{l+1}=\emptyset.$ However, the
following example demonstrates that the operators (17) and (18) are not the
same.

######  Example 4.2.

Put $\displaystyle X=[2],$ $\displaystyle f(1)=0.5,$ $\displaystyle f(2)=1,$
$\displaystyle\mu(X)=1,$ $\displaystyle\mu(\\{1\\})=0.5$ and
$\displaystyle\mu(\\{2\\})=0.4.$ By (16) with $\displaystyle\circ=\wedge$ and
$\displaystyle\widehat{\mu}=\mu$ we get

$\displaystyle\displaystyle{\mathfrak{C}}^{\mu,\mu}_{\wedge}(f)$
$\displaystyle\displaystyle=\sum_{i=1}^{2}f_{(i)}\wedge(\mu(B_{(i)})-\mu(B_{(i+1)}))$
$\displaystyle\displaystyle=0.5\wedge(\mu(X)-\mu(\\{2\\}))+1\wedge\mu(\\{2\\})=0.9.$

Considering $\displaystyle\mathrm{L}(x,y,z,w)=x\wedge(z-w)_{+}$ and the chain
$\displaystyle{\mathcal{D}}_{2}=\\{D_{2},D_{1}\\}$ with $\displaystyle
D_{2}=\\{1\\}$ and $\displaystyle D_{1}=X$ we obtain

$\displaystyle\displaystyle\mathrm{CS}^{\mathrm{L}}_{{\mathcal{H}}_{\text{chain}},{\mathscr{A}}^{\inf}}(f,\mu,\mu)$
$\displaystyle\displaystyle\geqslant\mathsf{A}^{\inf}(f|X)\wedge(\mu(X)-\mu(\\{1\\}))+\mathsf{A}^{\inf}(f|\\{1\\})\wedge\mu(\\{1\\})=1>{\mathfrak{C}}^{\mu,\mu}_{\wedge}(f).$

In consequence, operators (17) and (18) are different.

Reason for the non-equivalence between operators (17) and (18) is the lack of
symmetry of monotone measure. The following result provides a necessary and
sufficient condition under which both operators are equal.

######  Theorem 4.3.

Assume that $\displaystyle\circ\colon[0,\infty)^{2}\to[0,\infty)$ is
nondecreasing. Then the operator defined in (18) coincides with
$\displaystyle{\mathfrak{C}}^{\mu,\widehat{\mu}}_{\circ}(f)$ for any
$\displaystyle f\in\mathbf{F}$ and any symmetric
$\displaystyle\mu,\widehat{\mu}\in\mathbf{M}$ such that
$\displaystyle\mu\geqslant\widehat{\mu}$ if and only if the function
$\displaystyle x\mapsto a\circ x$ is subadditive for any $\displaystyle a.$

Proof. The value of
$\displaystyle{\mathfrak{C}}^{\mu,\widehat{\mu}}_{\circ}(f)$ for any
$\displaystyle f\in\mathbf{F}$ does not depend on permutations, since
$\displaystyle\mu,\widehat{\mu}$ are symmetric (see Proposition 4.1).
“$\displaystyle\Rightarrow$” Let $\displaystyle
f=b\mathbbm{1}_{\\{2,\ldots,n\\}},$ where $\displaystyle b\geqslant 0.$
Clearly,

$\displaystyle{\mathfrak{C}}^{\mu,\mu}_{\circ}(f)=0\circ(\mu(B_{(1)})-\mu(B_{(2)}))+\sum_{i=2}^{n}b\circ(\mu(B_{(i)})-\mu(B_{(i+1)}))$

for any symmetric $\displaystyle\mu\in\mathbf{M},$ where $\displaystyle
B_{(n+1)}=\emptyset.$ The operator (18) coincides with
$\displaystyle{\mathfrak{C}}^{\mu,\mu}_{\circ},$ so for the chain
$\displaystyle{\mathcal{D}}_{2}=\\{D_{2},D_{1}\\}$ with $\displaystyle
D_{2}=B_{(2)}$ and $\displaystyle D_{1}=B_{(1)}$ we get

$\displaystyle
0\circ(\mu(B_{(1)})-\mu(B_{(2)}))+b\circ\mu(B_{(2)})\leqslant{\mathfrak{C}}^{\mu,\mu}_{\circ}(f)$

for any symmetric $\displaystyle\mu\in\mathbf{M}.$ In consequence, we have

$\displaystyle\displaystyle
b\circ\sum_{i=2}^{n}(\mu(B_{(i)})-\mu(B_{(i+1)}))\leqslant\sum_{i=2}^{n}b\circ(\mu(B_{(i)})-\mu(B_{(i+1)}))$

for any symmetric $\displaystyle\mu\in\mathbf{M}.$ This implies that
$\displaystyle x\circ(y+z)\leqslant x\circ y+x\circ z$ for any $\displaystyle
x,y,z.$

“$\displaystyle\Leftarrow$” Since $\displaystyle\\{B_{(n)},\ldots,B_{(1)}\\}$
is a chain, by (17) we have

$\displaystyle\displaystyle\sup_{{\mathcal{D}}\in{\mathcal{H}}_{\text{chain}}}\sum_{i=1}^{l}\mathsf{A}^{\inf}(f|D_{i})\circ(\mu(D_{i})-\widehat{\mu}(D_{i+1}))\geqslant{\mathfrak{C}}^{\mu,\widehat{\mu}}_{\circ}(f).$

To prove the statement we will show that

$\displaystyle\displaystyle L$
$\displaystyle\displaystyle:=\sum_{i=1}^{l}\mathsf{A}^{\inf}(f|D_{i})\circ(\mu(D_{i})-\widehat{\mu}(D_{i+1}))\leqslant{\mathfrak{C}}^{\mu,\widehat{\mu}}_{\circ}(f)$

for any $\displaystyle{\mathcal{D}}_{l}\in{\mathcal{H}}_{\text{chain}}$ such
that $\displaystyle l\in[n].$ For the sake of clarity of presentation, we will
consider separately the case of a chain consisting of one nonempty set.

  * •

Let
$\displaystyle{\mathcal{D}}_{1}=\\{D_{1}\\}\in{\mathcal{H}}_{\text{chain}}.$
Then $\displaystyle L=\mathsf{A}^{\inf}(f|D_{1})\circ\mu(D_{1}),$ as
$\displaystyle D_{2}=\emptyset,$ due to the convention. Since $\displaystyle
X$ is a finite set, so
$\displaystyle\\{k\colon\mathsf{A}^{\inf}(f|D_{1})=f_{(k)}\\}\neq\emptyset.$
Let $\displaystyle k_{1}=\min\\{k\colon\mathsf{A}^{\inf}(f|D_{1})=f_{(k)}\\}.$
Thus $\displaystyle D_{1}\subseteq B_{(k_{1})}$ and by monotonicity of
$\displaystyle\circ$ and subadditivity of $\displaystyle x\mapsto a\circ x$ we
get

$\displaystyle\displaystyle L$ $\displaystyle\displaystyle\leqslant
f_{(k_{1})}\circ\mu(B_{(k_{1})})=f_{(k_{1})}\circ\big{(}\mu(B_{(k_{1})})-\widehat{\mu}(B_{(k_{1}+1)})+\widehat{\mu}(B_{(k_{1}+1)})\big{)}$
$\displaystyle\displaystyle\leqslant
f_{(k_{1})}\circ(\mu(B_{(k_{1})})-\widehat{\mu}(B_{(k_{1}+1)}))+f_{(k_{1})}\circ\widehat{\mu}(B_{(k_{1}+1)}).$

Since $\displaystyle(f_{(i)})_{i}$ is a nondecreasing sequence and
$\displaystyle\widehat{\mu}\leqslant\mu,$ we get

$\displaystyle\displaystyle L$ $\displaystyle\displaystyle\leqslant
f_{(k_{1})}\circ(\mu(B_{(k_{1})})-\widehat{\mu}(B_{(k_{1}+1)}))+f_{(k_{1}+1)}\circ\mu(B_{(k_{1}+1)})$
$\displaystyle\displaystyle\leqslant\ldots\leqslant\sum_{i=k_{1}}^{n}f_{(i)}\circ(\mu(B_{(i)})-\widehat{\mu}(B_{(i+1)}))\leqslant{\mathfrak{C}}^{\mu,\widehat{\mu}}_{\circ}(f).$

  * •

Let $\displaystyle{\mathcal{D}}_{l}\in{\mathcal{H}}_{\text{chain}}$ for fixed
$\displaystyle 2\leqslant l\leqslant n.$ Then $\displaystyle L=S_{1},$ where

$\displaystyle\displaystyle
S_{k}=\sum_{i=k}^{l}\mathsf{A}^{\inf}(f|D_{i})\circ(\mu(D_{i})-\widehat{\mu}(D_{i+1}))$

for $\displaystyle k\in[l]$ with the convention $\displaystyle
D_{l+1}=\emptyset.$ Let $\displaystyle
k_{1}=\min\\{k\colon\mathsf{A}^{\inf}(f|D_{1})=f_{(k)}\\}.$ Then
$\displaystyle D_{1}\subseteq B_{(k_{1})}$ and

$\displaystyle\displaystyle L$ $\displaystyle\displaystyle\leqslant
f_{(k_{1})}\circ(\mu(B_{(k_{1})})-\widehat{\mu}(D_{2}))+S_{2}.$ (19)

Note that $\displaystyle D_{2}\subset B_{(k_{1})}$ and
$\displaystyle|D_{2}|<|D_{1}|.$ Thus $\displaystyle|D_{2}|<|B_{(k_{1})}|$ and
$\displaystyle|D_{2}|\leqslant|B_{(k_{1}+1)}|,$ as
$\displaystyle|B_{(k_{1}+1)}|+1=|B_{(k_{1})}|.$ Then there exists
$\displaystyle k_{2}>k_{1}$ such that $\displaystyle|D_{2}|=|B_{(k_{2})}|$ and
$\displaystyle\mathsf{A}^{\inf}(f|D_{2})\leqslant\mathsf{A}^{\inf}(f|B_{(k_{2})})=f_{(k_{2})}.$
Indeed, if $\displaystyle D_{2}=B_{(k_{2})},$ then
$\displaystyle\mathsf{A}^{\inf}(f|D_{2})=\mathsf{A}^{\inf}(f|B_{(k_{2})}).$ If
$\displaystyle D_{2}\neq B_{(k_{2})},$ then there exists $\displaystyle j$
such that $\displaystyle j\in D_{2}$ and $\displaystyle j\notin B_{(k_{2})}$
as $\displaystyle|D_{2}|=|B_{(k_{2})}|.$ Observe that $\displaystyle
B_{(k_{1})}=\\{(k_{1}),\ldots,(k_{2}-1)\\}\cup B_{(k_{2})}$ and $\displaystyle
f(i)\leqslant\mathsf{A}^{\inf}(f|B_{(k_{2})})$ for any $\displaystyle
i\in\\{(k_{1}),\ldots,(k_{2}-1)\\},$ so $\displaystyle
f(j)\leqslant\mathsf{A}^{\inf}(f|B_{(k_{2})}),$ as $\displaystyle j\in
B_{(k_{1})}\setminus B_{(k_{2})}.$ In consequence
$\displaystyle\mathsf{A}^{\inf}(f|D_{2})\leqslant\mathsf{A}^{\inf}(f|B_{(k_{2})}).$
By (19) and the symmetricity of $\displaystyle\mu$ and
$\displaystyle\widehat{\mu},$ we have

$\displaystyle\displaystyle L$ $\displaystyle\displaystyle\leqslant
f_{(k_{1})}\circ(\mu(B_{(k_{1})})-\widehat{\mu}(B_{(k_{2})}))+f_{(k_{2})}\circ(\mu(B_{(k_{2})})-\widehat{\mu}(D_{3}))+S_{3}.$

Using $\displaystyle(k_{2}-k_{1}-1)$ times monotonicity of
$\displaystyle\circ$ and subadditivity of $\displaystyle x\mapsto a\circ x,$
we get

$\displaystyle\displaystyle L$
$\displaystyle\displaystyle\leqslant\sum_{i=k_{1}}^{k_{2}-1}f_{(i)}\circ(\mu(B_{(i)})-\widehat{\mu}(B_{(i+1)}))+f_{(k_{2})}\circ(\mu(B_{(k_{2})})-\widehat{\mu}(D_{3}))+S_{3}.$

Repeating the same arguments for $\displaystyle D_{3},\ldots,D_{l}$ leads to
the desired inequality

$\displaystyle\displaystyle L$
$\displaystyle\displaystyle\leqslant\sum_{i=k_{1}}^{k_{l}-1}f_{(i)}\circ(\mu(B_{(i)})-\widehat{\mu}(B_{(i+1)}))+f_{(k_{l})}\circ\mu(B_{(k_{l})})\leqslant{\mathfrak{C}}^{\mu,\widehat{\mu}}_{\circ}(f).$

This completes the proof. ∎

######  Remark 4.4.

The assumption on symmetricity of $\displaystyle\mu\in\mathbf{M}$ cannot be
omitted (see Example 4.2, where $\displaystyle x\mapsto a\wedge x$ is
subadditive for any $\displaystyle a\geqslant 0$).

In a similar way to the proof of Theorem 4.3 one can show that if
$\displaystyle\circ\colon[0,1]^{2}\to[0,1]$ is nondecreasing, then the
operator defined in (18) coincides with
$\displaystyle{\mathfrak{C}}^{\mu,\widehat{\mu}}_{\circ}(f)$ for any
$\displaystyle f\in\mathbf{F}^{1}$ and any symmetric capacities
$\displaystyle\mu,\widehat{\mu}$ such that
$\displaystyle\mu\geqslant\widehat{\mu}$ if and only if the function
$\displaystyle x\mapsto a\circ x$ is subadditive for any $\displaystyle
a\in[0,1].$ The symmetric capacities $\displaystyle\mu$ appear in the natural
way when dealing with the order statistics [16, 34] from discrete probability
distributions. This is due to the fact that the operator (16) with
$\displaystyle\circ=\cdot$ and symmetric capacity
$\displaystyle\mu=\widehat{\mu}$ is the Choquet integral being a
generalization of the OWA operator777R.R. Yager in [40] defined the ordered
weighted averaging (OWA) operator as an expected value of order statistics for
random variables defined in finite possible outcomes..

Now we show that the operator $\displaystyle{\mathfrak{C}}^{\mu,\mu}_{\circ}$
can be rewritten in terms of $\displaystyle t$-level sets.

######  Proposition 4.5.

If $\displaystyle a\circ b=g(a)b$ for some nonnegative function $\displaystyle
g$ with $\displaystyle g(0)=0,$ then

$\displaystyle\displaystyle{\mathfrak{C}}_{\circ}^{\mu,\mu}(f)=\sum_{i=1}^{n}f_{(i)}\circ\big{(}\mu(\\{f\geqslant
f_{(i)}\\})-\mu(\\{f\geqslant f_{(i+1)}\\})\big{)}$ (20)

holds for any $\displaystyle(f,\mu)\in\mathbf{F}\times\mathbf{M},$ where
$\displaystyle(\cdot)$ is a permutation such that $\displaystyle
f_{(1)}\leqslant\ldots\leqslant f_{(n)}$ under the convention
$\displaystyle\\{f\geqslant f_{(n+1)}\\}=\emptyset.$

Proof. If $\displaystyle(f_{(i)})_{i=1}^{n}$ is an increasing sequence, the
statement is obvious. Assume that $\displaystyle
f_{(1)}<\ldots<f_{(j)}=\ldots=f_{(k)}<f_{(k+1)}<\ldots<f_{(n)}$ for some
$\displaystyle j<k.$ Then by (16) and Abel transformation888Abel
transformation:
$\displaystyle\textstyle{\sum_{i=1}^{n}a_{i}(b_{i}-b_{i+1})=\sum_{i=1}^{n}b_{i}(a_{i}-a_{i-1})}$
under the convention $\displaystyle b_{n+1}=a_{0}=0.$ we get

$\displaystyle\displaystyle{\mathfrak{C}}_{\circ}^{\mu,\mu}(f)$
$\displaystyle\displaystyle=\sum_{i=1}^{n}(g(f_{(i)})-g(f_{(i-1)}))\mu(B_{(i)}),$

where $\displaystyle B_{(n+1)}=\emptyset,$ $\displaystyle g(0)=0$ and
$\displaystyle f_{(0)}=0.$ Moreover, $\displaystyle B_{(i)}=\\{f\geqslant
f_{(i)}\\}$ for any $\displaystyle i\notin\\{j+1,\ldots,k\\}.$ As
$\displaystyle g(f_{(j)})=\ldots=g(f_{(k)}),$ we get

$\displaystyle\displaystyle{\mathfrak{C}}_{\circ}^{\mu,\mu}(f)$
$\displaystyle\displaystyle=\sum_{i=1}^{j}(g(f_{(i)})-g(f_{(i-1)}))\mu(\\{f\geqslant
f_{(i)}\\})+\sum_{i=k+1}^{n}(g(f_{(i)})-g(f_{(i-1)}))\mu(\\{f\geqslant
f_{(i)}\\})$
$\displaystyle\displaystyle=\sum_{i=1}^{n}(g(f_{(i)})-g(f_{(i-1)}))\mu(\\{f\geqslant
f_{(i)}\\})=\sum_{i=1}^{n}g(f_{(i)})\big{(}\mu(\\{f\geqslant
f_{(i)}\\})-\mu(\\{f\geqslant f_{(i+1)}\\})\big{)},$

where in the last equality the Abel transformation has been used again. The
other cases can be treated similarly, so we omit them. ∎

It is clear that (20) is true for any $\displaystyle\mu\in\mathbf{M}$ and any
binary operation $\displaystyle\circ$ whenever $\displaystyle f\in\mathbf{F}$
is decreasing or increasing. However, it is not possible to obtain (20) for
any $\displaystyle f\in\mathbf{F}$ and any binary operation
$\displaystyle\circ$ even if $\displaystyle\mu\in\mathbf{M}$ is symmetric as
it is illustrated in the following example.

######  Example 4.6.

Let $\displaystyle X=[2],$ $\displaystyle f(1)=f(2)=0{.}5$ and
$\displaystyle\mu\in\mathbf{M}$ be symmetric such that $\displaystyle\mu(X)=2$
and $\displaystyle\mu(\\{2\\})=1=\mu(\\{1\\}).$ The operator
$\displaystyle{\mathfrak{C}}_{\wedge}^{\mu,\mu}$ does not depend on
permutation, so for $\displaystyle B_{(1)}=X$ and $\displaystyle|B_{(2)}|=1$
we get

$\displaystyle\displaystyle{\mathfrak{C}}_{\wedge}^{\mu,\mu}(f)=0{.}5\wedge(\mu(X)-\mu(B_{(2)}))+0{.}5\wedge\mu(B_{(2)})=1.$

The right-hand side of (20) with $\displaystyle\circ=\wedge$ takes the form

$\displaystyle\displaystyle\sum_{i=1}^{2}f_{(i)}\wedge\big{(}\mu(\\{f\geqslant
f_{(i)}\\})-\mu(\\{f\geqslant f_{(i+1)}\\})\big{)}=0{.}5\wedge 0+0{.}5\wedge
2=0{.}5<{\mathfrak{C}}_{\wedge}^{\mu,\mu}(f).$

To sum up, the equality (20) does not hold.

### Connection with
$\displaystyle{\mathfrak{R}}{\mathfrak{C}}^{\mu,\widehat{\mu}}_{\circ}$

Let $\displaystyle\circ\colon[0,\infty)^{2}\to[0,\infty).$ Define the operator
$\displaystyle{\mathfrak{R}}{\mathfrak{C}}^{\mu,\widehat{\mu}}_{\circ}$ as
follows

$\displaystyle\displaystyle{\mathfrak{R}}{\mathfrak{C}}^{\mu,\widehat{\mu}}_{\circ}(f)=\sum_{i=1}^{n}f_{(i)}\circ(\mu(C_{(i)})-\widehat{\mu}(C_{(i-1)}))$
(21)

for any
$\displaystyle(f,\mu,\widehat{\mu})\in\mathbf{F}\times\mathbf{M}\times\mathbf{M}$
such that $\displaystyle\mu\geqslant\widehat{\mu},$ where
$\displaystyle(\cdot)\colon[n]\to[n]$ is a permutation such that
$\displaystyle f_{(1)}\leqslant\ldots\leqslant f_{(n)}$ and $\displaystyle
C_{(i)}=\\{(1),\ldots,(i)\\}$ for $\displaystyle i\in[n]$ with $\displaystyle
C_{(0)}=\emptyset.$ The operator (21) is well defined if $\displaystyle f$ is
monotone (decreasing or increasing),
$\displaystyle\mu,\widehat{\mu}\in\mathbf{M}$ are symmetric or $\displaystyle
a\circ b=g(a)b$ for some $\displaystyle g.$ Putting
$\displaystyle\mu=\widehat{\mu}\in\mathbf{M}^{1}$ and
$\displaystyle\circ=\cdot$ in (21), we obtain the reverse Choquet integral
introduced in [29].

Observe that if $\displaystyle\mu([n])=\widehat{\mu}([n]),$
$\displaystyle\nu(D)=\widehat{\mu}([n])-\widehat{\mu}(D^{c})$ and
$\displaystyle\widehat{\nu}(D)=\mu([n])-\mu(D^{c})$ for any $\displaystyle
D\in 2^{[n]},$ then

$\displaystyle\displaystyle{\mathfrak{R}}{\mathfrak{C}}^{\mu,\widehat{\mu}}_{\circ}(f)$
$\displaystyle\displaystyle={\mathfrak{C}}^{\nu,\widehat{\nu}}_{\circ}(f).$
(22)

Hence, Theorem 4.3 can also be used to analyze the relationship between the
special case of Choquet-Sugeno-like operator and
$\displaystyle{\mathfrak{R}}{\mathfrak{C}}_{\circ}^{\mu,\widehat{\mu}}.$

### Connection with $\displaystyle{\mathfrak{C}}^{\circ,\delta}_{\mu}$

######  Definition 4.7.

A function $\displaystyle\delta\colon[0,\infty)^{2}\to[0,\infty)$ is said to
be a dissimilarity function if for all $\displaystyle x,y,z$ the following
conditions hold:

  1. (a)

$\displaystyle\delta(x,y)=\delta(y,x)$;

  2. (b)

$\displaystyle\delta(x,y)=0$ if and only if $\displaystyle x=y$;

  3. (c)

if $\displaystyle x\leqslant y\leqslant z,$ then
$\displaystyle\delta(x,y)\leqslant\delta(x,z)$ and
$\displaystyle\delta(y,z)\leqslant\delta(x,z).$

Let $\displaystyle\circ\colon[0,\infty)^{2}\to[0,\infty)$ and
$\displaystyle\delta$ be a dissimilarity function. Now we define the following
operator

$\displaystyle\displaystyle{\mathfrak{C}}_{\mu}^{\circ,\delta}(f)=\sum_{i=1}^{n}\delta(f_{(i)},f_{(i-1)})\circ\mu(B_{(i)})$
(23)

for any $\displaystyle(f,\mu)\in\mathbf{F}\times\mathbf{M},$ where
$\displaystyle(\cdot)\colon[n]\to[n]$ is a permutation such that
$\displaystyle f_{(1)}\leqslant\ldots\leqslant f_{(n)},$ $\displaystyle
B_{(i)}=\\{(i),\ldots,(n)\\}$ for $\displaystyle i\in[n]$ under the convention
$\displaystyle f_{(0)}=0.$ In order the operator (23) would not depend on the
permutation it is necessary to assume that either $\displaystyle 0\circ b=0$
for all $\displaystyle b$ (cf. [30]) or $\displaystyle f\in\mathbf{F}$ is
monotone (decreasing or increasing) or $\displaystyle\mu\in\mathbf{M}$ is
symmetric.

The operator (23) with $\displaystyle\circ=\cdot,$
$\displaystyle(f,\mu)\in\mathbf{F}^{1}\times\mathbf{M}^{1}$ and
$\displaystyle\delta\colon[0,1]^{2}\to[0,1]$ being a restricted dissimilarity
function999 A function $\displaystyle\delta\colon[0,1]^{2}\to[0,1]$ is said to
be a restricted dissimilarity function if it satisfies, for all $\displaystyle
x,y,z\in[0,1],$ the conditions (a), (b) and (c) from Definition 4.7 and
$\displaystyle\delta(x,y)=1$ if and only if $\displaystyle\\{x,y\\}=\\{0,1\\}$
(see [8, Definition 2.1]). is the $\displaystyle d$-Choquet integral defined
in [8]. Study of (23) for $\displaystyle\circ\colon[0,1]^{2}\to[0,1],$
$\displaystyle(f,\mu)\in\mathbf{F}^{1}\times\mathbf{M}^{1}$ and
$\displaystyle\delta(x,y)=|x-y|,$ i.e.,

$\displaystyle\displaystyle{\mathfrak{C}}_{\mu}^{\circ}(f)=\sum_{i=1}^{n}(f_{(i)}-f_{(i-1)})\circ\mu(B_{(i)}),$

was conducted by Mesiar et al. [30] and call it as $\displaystyle\circ$-based
discrete Choquet-like integral. Later, it was redefined by Lucca et al. [27]
in the way
$\displaystyle\overline{{\mathfrak{C}}}_{\mu}^{\circ}(f)=\min\\{1,{\mathfrak{C}}_{\mu}^{\circ}(f)\\}$
for any $\displaystyle(f,\mu)\in\mathbf{F}^{1}\times\mathbf{M}^{1}$ and was
used in a fuzzy rule-based classification system.

Similarly as earlier, we give a relationship between (23) and the Choquet-
Sugeno-like operator (5) with
$\displaystyle{\mathcal{H}}={\mathcal{H}}_{\text{chain}},$
$\displaystyle{\mathcal{R}}={\mathcal{R}}^{-},$
$\displaystyle\mathrm{L}(x,y,z,w)=\delta(x,y)\circ z$ and
$\displaystyle{\mathscr{A}}={\mathscr{A}}^{\inf}=\widehat{{\mathscr{A}}},$
i.e.,

$\displaystyle\displaystyle\mathrm{CS}^{\mathrm{L}}_{{\mathcal{H}}_{\text{chain}},{\mathscr{A}}^{\inf},{\mathscr{A}}^{\inf}}(f,\mu)=\sup_{{\mathcal{D}}\in{\mathcal{H}}_{\text{chain}}}\sum_{i=1}^{l}\delta\big{(}\mathsf{A}^{\inf}(f|D_{i}),\mathsf{A}^{\inf}(f|D_{i-1})\big{)}\circ\mu(D_{i})$
(24)

for any $\displaystyle(f,\mu)\in\mathbf{F}\times\mathbf{M}$ under the
convention $\displaystyle D_{0}=\emptyset,$ where
$\displaystyle\circ\colon[0,\infty)^{2}\to[0,\infty)$ and
$\displaystyle\delta$ is a dissimilarity function. We split the study of
connection between $\displaystyle{\mathfrak{C}}^{\circ,\delta}_{\mu}$ and (24)
into two cases: $\displaystyle|X|=2$ and $\displaystyle|X|\geqslant 3.$

######  Theorem 4.8.

Let $\displaystyle X=[2].$ Assume that $\displaystyle\delta$ is a
dissimilarity function and
$\displaystyle\circ\colon[0,\infty)^{2}\to[0,\infty)$ is nondecreasing.

  * (i)

The operator defined in (24) coincides with
$\displaystyle{\mathfrak{C}}_{\mu}^{\circ,\delta}(f)$ for any $\displaystyle
f\in\mathbf{F}$ and any symmetric $\displaystyle\mu\in\mathbf{M}$ if and only
if

$\displaystyle\displaystyle\delta(x_{2},0)\circ y\leqslant\delta(x_{1},0)\circ
y+\delta(x_{2},x_{1})\circ y$ (25)

for any $\displaystyle 0\leqslant x_{1}\leqslant x_{2}<\infty$ and any
$\displaystyle y\in[0,\infty).$

  * (ii)

If $\displaystyle 0\circ a=0$ for all $\displaystyle a,$ then the operator
defined in (24) coincides with
$\displaystyle{\mathfrak{C}}_{\mu}^{\circ,\delta}(f)$ for any
$\displaystyle(f,\mu)\in\mathbf{F}\times\mathbf{M}$ if and only if (25) is
true for any $\displaystyle 0\leqslant x_{1}\leqslant x_{2}<\infty$ and any
$\displaystyle y\in[0,\infty).$

Proof. (i) Since $\displaystyle X=[2],$ so
$\displaystyle{\mathcal{H}}_{\text{chain}}$ is given in Example 3.1 (c). Let
$\displaystyle x_{(1)}=\min\\{x_{1},x_{2}\\}$ and $\displaystyle
x_{(2)}=\max\\{x_{1},x_{2}\\}.$ Then the operators take the form

$\displaystyle\displaystyle\mathrm{CS}^{\mathrm{L}}_{{\mathcal{H}}_{\text{chain}},{\mathscr{A}}^{\inf},{\mathscr{A}}^{\inf}}(f,\mu)$
$\displaystyle\displaystyle=\max\big{\\{}\delta(x_{(2)},0)\circ\mu(\\{(2)\\}),\,\delta(x_{(1)},0)\circ\mu(X)+\delta(x_{(2)},x_{(1)})\circ\mu(\\{(2)\\})\big{\\}},$
(26) $\displaystyle\displaystyle{\mathfrak{C}}_{\mu}^{\circ,\delta}(f)$
$\displaystyle\displaystyle=\delta(x_{(1)},0)\circ\mu(X)+\delta(x_{(2)},x_{(1)})\circ\mu(\\{(2)\\})$

for any symmetric $\displaystyle\mu\in\mathbf{M}.$ Thus, the equality
$\displaystyle\mathrm{CS}^{\mathrm{L}}_{{\mathcal{H}}_{\text{chain}},{\mathscr{A}}^{\inf},{\mathscr{A}}^{\inf}}(f,\mu)={\mathfrak{C}}_{\mu}^{\circ,\delta}(f)$
holds for any $\displaystyle f\in\mathbf{F}$ and any symmetric
$\displaystyle\mu\in\mathbf{M}$ if and only if

$\displaystyle\displaystyle\delta(x_{(2)},0)\circ\mu(\\{(2)\\})\leqslant\delta(x_{(1)},0)\circ\mu(X)+\delta(x_{(2)},x_{(1)})\circ\mu(\\{(2)\\})$

for any $\displaystyle 0\leqslant x_{1}\leqslant x_{2}<\infty$ and any
symmetric $\displaystyle\mu.$ As $\displaystyle\mu(\\{2\\})\leqslant\mu(X)$
and $\displaystyle\circ$ is nondecreasing, the latter condition is equivalent
to $\displaystyle\delta(x_{(2)},0)\circ y\leqslant\delta(x_{(1)},0)\circ
y+\delta(x_{(2)},x_{(1)})\circ y$ for any $\displaystyle 0\leqslant
x_{1}\leqslant x_{2}<\infty$ and any $\displaystyle 0\leqslant y<\infty,$
which finishes the proof of part (i).

The proof of part (ii) is similar to the proof of (i), since (26) holds for
any $\displaystyle\mu\in\mathbf{M}$ in view of $\displaystyle 0\circ a=0$ for
any $\displaystyle a.$ ∎

######  Theorem 4.9.

Let $\displaystyle|X|\geqslant 3.$ Assume that $\displaystyle\delta$ is a
dissimilarity function and
$\displaystyle\circ\colon[0,\infty)^{2}\to[0,\infty)$ is nondecreasing such
that $\displaystyle 0\circ 0=0.$ Then the operator defined in (24) coincides
with $\displaystyle{\mathfrak{C}}_{\mu}^{\circ,\delta}(f)$ for any
$\displaystyle f\in\mathbf{F}$ and any symmetric
$\displaystyle\mu\in\mathbf{M}$ if and only if
$\displaystyle\delta(x_{3},x_{1})\circ y\leqslant\delta(x_{2},x_{1})\circ
y+\delta(x_{3},x_{2})\circ y$ for any $\displaystyle 0\leqslant x_{1}\leqslant
x_{2}\leqslant x_{3}<\infty$ and any $\displaystyle y\in[0,\infty).$

Proof. “$\displaystyle\Rightarrow$” Assume that $\displaystyle f(1)=x_{1},$
$\displaystyle f(2)=x_{2}$ and $\displaystyle f(j)=x_{3}$ for $\displaystyle
j\in\\{3,\ldots,n\\}$ such that $\displaystyle 0\leqslant x_{1}\leqslant
x_{2}\leqslant x_{3},$ where $\displaystyle n=|X|.$ Clearly,

$\displaystyle{\mathfrak{C}}_{\mu}^{\circ,\delta}(f)=\delta(x_{1},0)\circ\mu(B_{(1)})+\delta(x_{2},x_{1})\circ\mu(B_{(2)})+\delta(x_{3},x_{2})\circ\mu(B_{(3)})+\sum_{i=4}^{n}0\circ\mu(B_{(i)}).$

The operator (24) coincides with
$\displaystyle{\mathfrak{C}}_{\mu}^{\circ,\delta},$ so for
$\displaystyle{\mathcal{D}}_{2}=\\{B_{(3)},B_{(1)}\\}\in{\mathcal{H}}_{\text{chain}},$
we get

$\displaystyle\delta(x_{1},0)\circ\mu(B_{(1)})+\delta(x_{3},x_{1})\circ\mu(B_{(3)})\leqslant{\mathfrak{C}}_{\mu}^{\circ,\delta}(f)$

for any symmetric $\displaystyle\mu\in\mathbf{M}.$ In consequence, for any
$\displaystyle y\in[0,\infty)$ and the symmetric monotone measure given by

$\displaystyle\mu(E)=\begin{cases}y,&\textrm{if}\,|E|\in\\{n,n-1,n-2\\},\\\
0,&\textrm{otherwise},\end{cases}$

we obtain $\displaystyle\delta(x_{3},x_{1})\circ
y\leqslant\delta(x_{2},x_{1})\circ y+\delta(x_{3},x_{2})\circ y,$ as
$\displaystyle 0\circ 0=0,$ which finishes the proof of the subadditivity
condition.

“$\displaystyle\Leftarrow$” Since
$\displaystyle\mathsf{A}^{\inf}(f|B_{(i)})=f_{(i)}$ for any $\displaystyle
i\in[n],$ we get

$\displaystyle\sup_{{\mathcal{D}}\in{\mathcal{H}}_{\text{chain}}}\sum_{i=1}^{l}\delta\big{(}\mathsf{A}^{\inf}(f|D_{i}),\mathsf{A}^{\inf}(f|D_{i-1})\big{)}\circ\mu(D_{i})\geqslant{\mathfrak{C}}_{\mu}^{\circ,\delta}(f),$

so it is enough to show that

$\displaystyle\displaystyle
L:=\sum_{i=1}^{l}\delta\big{(}\mathsf{A}^{\inf}(f|D_{i}),\mathsf{A}^{\inf}(f|D_{i-1})\big{)}\circ\mu(D_{i})\leqslant{\mathfrak{C}}_{\mu}^{\circ,\delta}(f)$

for any $\displaystyle{\mathcal{D}}_{l}\in{\mathcal{H}}_{\text{chain}}$ such
that $\displaystyle l\in[n].$ Then $\displaystyle L=S_{1},$ where

$\displaystyle\displaystyle
S_{k}=\sum_{i=k}^{l}\delta\big{(}\mathsf{A}^{\inf}(f|D_{i}),\mathsf{A}^{\inf}(f|D_{i-1})\big{)}\circ\mu(D_{i})$

for $\displaystyle k\in[l].$ Let $\displaystyle
k_{1}=\min\\{k\in[n]\colon\mathsf{A}^{\inf}(f|D_{1})=f_{(k)}\\}.$ Thus
$\displaystyle D_{1}\subseteq B_{(k_{1})}$ and

$\displaystyle\displaystyle
L\leqslant\delta(f_{(k_{1})},0)\circ\mu(B_{(k_{1})})+\delta(\mathsf{A}^{\inf}(f|D_{2}),f_{(k_{1})})\circ\mu(D_{2})+S_{3}.$
(27)

Then there exists $\displaystyle k_{2}\in\\{k_{1}+1,\ldots,n\\}$ such that
$\displaystyle|D_{2}|=|B_{(k_{2})}|$ and
$\displaystyle\mathsf{A}^{\inf}(f|D_{2})\leqslant\mathsf{A}^{\inf}(f|B_{(k_{2})})=f_{(k_{2})}$
(see the proof of Theorem 4.3). In view of $\displaystyle
f_{(k_{1})}=\mathsf{A}^{\inf}(f|D_{1})\leqslant\mathsf{A}^{\inf}(f|D_{2})$ (as
$\displaystyle D_{2}\subset D_{1}$) and Definition 4.7 (a) and (c), from (27)
and symmetricity of $\displaystyle\mu$ we get

$\displaystyle\displaystyle L$
$\displaystyle\displaystyle\leqslant\delta(f_{(k_{1})},0)\circ\mu(B_{(k_{1})})+\delta(f_{(k_{2})},f_{(k_{1})})\circ\mu(B_{(k_{2})})+\delta(\mathsf{A}^{\inf}(f|D_{3}),f_{(k_{2})})\circ\mu(D_{3})+S_{4}.$
(28)

Now we only focus on $\displaystyle\delta(f_{(k_{1})},0)\circ\mu(B_{(k_{1})})$
from (28). By the assumption on $\displaystyle\delta$ and by monotonicity of
$\displaystyle\mu$ we obtain

$\displaystyle\displaystyle\delta(f_{(k_{1})},0)\circ\mu(B_{(k_{1})})$
$\displaystyle\displaystyle\leqslant\delta(f_{(1)},f_{(0)})\circ\mu(B_{(k_{1})})+\delta(f_{(k_{1})},f_{(1)})\circ\mu(B_{(k_{1})})$
$\displaystyle\displaystyle\leqslant\delta(f_{(1)},f_{(0)})\circ\mu(B_{(1)})+\delta(f_{(k_{1})},f_{(1)})\circ\mu(B_{(k_{1})})$
$\displaystyle\displaystyle\leqslant\ldots\leqslant\sum_{i=1}^{k_{1}}\delta(f_{(i)},f_{(i-1)})\circ\mu(B_{(i)}).$
(29)

Now consider the term
$\displaystyle\delta(f_{(k_{2})},f_{(k_{1})})\circ\mu(B_{(k_{2})})$ from (28).
Using the similar argument as above we get

$\displaystyle\displaystyle\delta(f_{(k_{2})},f_{(k_{1})})\circ\mu(B_{(k_{2})})$
$\displaystyle\displaystyle\leqslant\delta(f_{(k_{1}+1)},f_{(k_{1})})\circ\mu(B_{(k_{2})})+\delta(f_{(k_{2})},f_{(k_{1}+1)})\circ\mu(B_{(k_{2})})$
$\displaystyle\displaystyle\leqslant\delta(f_{(k_{1}+1)},f_{(k_{1})})\circ\mu(B_{(k_{1}+1)})+\delta(f_{(k_{2})},f_{(k_{1}+1)})\circ\mu(B_{(k_{2})})$
$\displaystyle\displaystyle\leqslant\ldots\leqslant\sum_{i=k_{1}+1}^{k_{2}}\delta(f_{(i)},f_{(i-1)})\circ\mu(B_{(i)}).$
(30)

So putting (4)–(4) in (28) we have

$\displaystyle\displaystyle L$
$\displaystyle\displaystyle\leqslant\sum_{i=1}^{k_{2}}\delta(f_{(i)},f_{(i-1)})\circ\mu(B_{(i)})+\delta(\mathsf{A}^{\inf}(f|D_{3}),f_{(k_{2})})\circ\mu(D_{3})+S_{4}.$

Repeating the same arguments for $\displaystyle D_{3},\ldots,D_{l}$ leads to
the inequality

$\displaystyle\displaystyle
L\leqslant\sum_{i=1}^{k_{l}}\delta(f_{(i)},f_{(i-1)})\circ\mu(B_{(i)})\leqslant{\mathfrak{C}}_{\mu}^{\circ,\delta}(f),$

which finishes the proof. ∎

######  Remark 4.10.

Note that the assumption on $\displaystyle|X|\geqslant 3$ and $\displaystyle
0\circ 0=0$ is used only in the implication “$\displaystyle\Rightarrow$” in
the proof of Theorem 4.9.

From Theorem 4.13 the two results connecting our approach with the existing
ones in the literature for the set $\displaystyle X$ such that
$\displaystyle|X|\geqslant 3$ follow:

  1. (a)

if $\displaystyle\delta\colon[0,1]^{2}\to[0,1]$ is a restricted dissimilarity
function, then the Choquet-Sugeno-like operator defined in (24) with
$\displaystyle\circ=\cdot$ coincides with the $\displaystyle d$-Choquet
integral for any $\displaystyle f\in\mathbf{F}^{1}$ and any symmetric capacity
$\displaystyle\mu$ if and only if
$\displaystyle\delta(x_{3},x_{1})\leqslant\delta(x_{2},x_{1})+\delta(x_{3},x_{2})$
for any $\displaystyle 0\leqslant x_{1}\leqslant x_{2}\leqslant x_{3}\leqslant
1.$

  2. (b)

if $\displaystyle\delta(x,y)=|x-y|,$ an operation
$\displaystyle\circ\colon[0,1]^{2}\to[0,1]$ is nondecreasing and
$\displaystyle 0\circ 0=0,$ then the operator (24) coincides with
$\displaystyle{\mathfrak{C}}_{\mu}^{\circ}(f)$ for any $\displaystyle
f\in\mathbf{F}^{1}$ and any symmetric capacity $\displaystyle\mu$ if and only
if $\displaystyle x\mapsto x\circ b$ is subadditive for any $\displaystyle
b\in[0,1].$

A representation of $\displaystyle{\mathfrak{C}}_{\mu}^{\circ,\delta}$ using
$\displaystyle t$-level sets reads as follows. The result is an easy
consequence of the fact that $\displaystyle\delta(x,x)=0$ for any
$\displaystyle x.$

######  Proposition 4.11.

Let $\displaystyle\delta$ be a dissimilarity function and
$\displaystyle\circ\colon[0,\infty)^{2}\to[0,\infty).$ If $\displaystyle
0\circ a=0$ for all $\displaystyle a,$ then

$\displaystyle\displaystyle{\mathfrak{C}}^{\circ,\delta}_{\mu}(f)=\sum_{i=1}^{n}\delta(f_{(i)},f_{(i-1)})\circ\mu(\\{f\geqslant
f_{(i)}\\})$ (31)

for any $\displaystyle(f,\mu)\in\mathbf{F}\times\mathbf{M},$ where
$\displaystyle(\cdot)\colon[n]\to[n]$ is a permutation such that
$\displaystyle f_{(1)}\leqslant\ldots\leqslant f_{(n)}$ with $\displaystyle
f_{(0)}=0.$

It is clear that (31) is true for any $\displaystyle\mu\in\mathbf{M}$ and any
binary operation $\displaystyle\circ$ whenever $\displaystyle f\in\mathbf{F}$
is decreasing or increasing. Again, it is not possible to obtain the equality
(31) for any $\displaystyle f\in\mathbf{F}$ and any binary operation
$\displaystyle\circ$ even for symmetric $\displaystyle\mu\in\mathbf{M}.$

######  Example 4.12.

Let $\displaystyle X,$ $\displaystyle f$ and $\displaystyle\mu$ be the same as
in Example 4.6. Then

$\displaystyle\displaystyle\sum_{i=1}^{2}\delta(f_{(i)},f_{(i-1)})\circ\mu(\\{f\geqslant
f_{(i)}\\})$
$\displaystyle\displaystyle=\delta(f_{(1)},0)\circ\mu(X)+0\circ\mu(X),$
$\displaystyle\displaystyle\sum_{i=1}^{2}\delta(f_{(i)},f_{(i-1)})\circ\mu(B_{(i)})$
$\displaystyle\displaystyle=\delta(f_{(1)},0)\circ\mu(X)+0\circ\mu(\\{2\\}).$

Clearly, for $\displaystyle 0\circ\mu(X)\neq 0\circ\mu(\\{2\\})$ the equality
in (31) does not hold.

### Connection with
$\displaystyle{\mathfrak{C}}_{\mu}^{(\mathrm{F}_{1},\mathrm{F}_{2})}$

Let
$\displaystyle\mathrm{F}_{1},\mathrm{F}_{2}\colon[0,\infty)^{2}\to[0,\infty)$
and $\displaystyle\mathrm{F}_{1}\geqslant\mathrm{F}_{2}.$ The operator
$\displaystyle{\mathfrak{C}}_{\mu}^{(\mathrm{F}_{1},\mathrm{F}_{2})}$ is
defined as

$\displaystyle\displaystyle{\mathfrak{C}}_{\mu}^{(\mathrm{F}_{1},\mathrm{F}_{2})}(f)$
$\displaystyle\displaystyle=\sum_{i=1}^{n}\big{(}\mathrm{F}_{1}(f_{(i)},\mu(B_{(i)}))-\mathrm{F}_{2}(f_{(i-1)},\mu(B_{(i)}))\big{)},$
(32)

where $\displaystyle(\cdot)\colon[n]\to[n]$ is a permutation such that
$\displaystyle f_{(1)}\leqslant\ldots\leqslant f_{(n)}$ and $\displaystyle
B_{(i)}=\\{(i),\ldots,(n)\\}$ for $\displaystyle i\in[n]$ with $\displaystyle
f_{(0)}=0.$ The formula (32) can be rewritten as follows

$\displaystyle\displaystyle{\mathfrak{C}}_{\mu}^{(\mathrm{F}_{1},\mathrm{F}_{2})}(f)=\sum_{i=1}^{n-1}\big{(}\mathrm{F}_{1}(f_{(i)},\mu(B_{(i)}))-\mathrm{F}_{2}(f_{(i)},\mu(B_{(i+1)}))\big{)}+\mathrm{F}_{1}(f_{(n)},\mu(B_{(n)}))-\mathrm{F}_{2}(0,\mu(B_{(1)})).$
(33)

The operator (32) is well defined (i.e., independent on permutations) when
either $\displaystyle\mathrm{F}_{1}=\mathrm{F}_{2}$ or $\displaystyle
f\in\mathbf{F}$ is monotone (decreasing or increasing) or
$\displaystyle\mu\in\mathbf{M}$ is symmetric.

Putting
$\displaystyle\mathrm{F}{{}_{1}}|_{[0,1]^{2}}=\mathrm{F}{{}_{2}}|_{[0,1]^{2}}=\mathrm{C}$
in (32), where $\displaystyle\mathrm{C}$ is a copula, we get the operator
firstly defined in [23] and then redefined in [26] under the name
$\displaystyle\mathrm{C}\mathrm{C}$-integral (or Choquet-like Copula-based
integral) of $\displaystyle(f,\mu)\in\mathbf{F}^{1}\times\mathbf{M}^{1}$ in
the form

$\displaystyle\displaystyle{\mathfrak{C}}_{\mu}^{(\mathrm{C},\mathrm{C})}(f)=\sum_{i=1}^{n}\big{(}\mathrm{C}(f_{(i)},\mu(B_{(i)}))-\mathrm{C}(f_{(i-1)},\mu(B_{(i)}))\big{)},$
(34)

where $\displaystyle f_{(i)}$ and $\displaystyle B_{(i)}$ are the same as in
(32). The operator (34) is a $\displaystyle[0,1]$-valued universal integral
([23]). Setting
$\displaystyle\mathrm{F}_{1},\mathrm{F}_{2}\colon[0,1]^{2}\to[0,1]$ in (32)
such that $\displaystyle\mathrm{F}_{1}\geqslant\mathrm{F}_{2}$ and
$\displaystyle\mathrm{F}_{1}(a,\mu(X))=a$ for all $\displaystyle a,$ we get

$\displaystyle\displaystyle{\mathfrak{C}}_{\mu}^{(\mathrm{F}_{1},\mathrm{F}_{2})}(f)=f_{(1)}+\sum_{i=2}^{n}\big{(}\mathrm{F}_{1}(f_{(i)},\mu(B_{(i)}))-\mathrm{F}_{2}(f_{(i-1)},\mu(B_{(i)}))\big{)}$

for any $\displaystyle(f,\mu)\in\mathbf{F}^{1}\times\mathbf{M}^{1},$ being a
version of $\displaystyle C_{\mathrm{F}_{1}\mathrm{F}_{2}}$-integral defined
in [28, Definition 7].

Let $\displaystyle{\mathcal{H}}={\mathcal{H}}_{\text{chain}}$ and
$\displaystyle{\mathcal{R}}={\mathcal{R}}^{-}.$ Then the Choquet-Sugeno-like
operator (5) with
$\displaystyle\mathrm{L}(x,y,z,w)=\mathrm{F}_{1}(x,z)-\mathrm{F}_{2}(y,z)$ and
$\displaystyle{\mathscr{A}}={\mathscr{A}}^{\inf}=\widehat{{\mathscr{A}}}$ can
be written as follows

$\displaystyle\displaystyle\mathrm{CS}^{(\mathrm{F}_{1},\mathrm{F}_{2})}_{{\mathcal{H}}_{\text{chain}},{\mathscr{A}}^{\inf},{\mathscr{A}}^{\inf}}(f,\mu)=\sup_{{\mathcal{D}}\in{\mathcal{H}}_{\text{chain}}}\sum_{i=1}^{l}\big{(}\mathrm{F}_{1}(\mathsf{A}^{\inf}(f|D_{i}),\mu(D_{i}))-\mathrm{F}_{2}(\mathsf{A}^{\inf}(f|D_{i-1}),\mu(D_{i}))\big{)}$
(35)

under the convention $\displaystyle D_{0}=\emptyset.$

A pair $\displaystyle(\mathrm{F}_{1},\mathrm{F}_{2})$ of functions
$\displaystyle\mathrm{F}_{1},\mathrm{F}_{2}\colon E_{1}\times
E_{2}\to[0,\infty),$ where $\displaystyle E_{1},E_{2}\subseteq[0,\infty),$ is
pairwise $\displaystyle 2$-increasing if

$\displaystyle\displaystyle\mathrm{F}_{1}(x_{1},y_{2})-\mathrm{F}_{2}(x_{1},y_{1})\leqslant\mathrm{F}_{1}(x_{2},y_{2})-\mathrm{F}_{2}(x_{2},y_{1})$

for any $\displaystyle[x_{1},x_{2}]\times[y_{1},y_{2}]\subseteq E_{1}\times
E_{2}.$ It is clear that the pair $\displaystyle(\mathrm{C},\mathrm{C})$ is
pairwise $\displaystyle 2$-increasing whenever $\displaystyle\mathrm{C}$ is a
copula [32].

######  Theorem 4.13.

Assume that
$\displaystyle\mathrm{F}_{1},\mathrm{F}_{2}\colon[0,\infty)^{2}\to[0,\infty)$
are pairwise $\displaystyle 2$-increasing, $\displaystyle\mathrm{F}_{1}$ is
nondecreasing, $\displaystyle\mathrm{F}_{1}\geqslant\mathrm{F}_{2}$ and
$\displaystyle\mathrm{F}_{2}(0,b)=0$ for any $\displaystyle b.$ Then the
operator (35) coincides with
$\displaystyle{\mathfrak{C}}_{\mu}^{(\mathrm{F}_{1},\mathrm{F}_{2})}(f)$ for
any $\displaystyle f\in\mathbf{F}$ and any symmetric
$\displaystyle\mu\in\mathbf{M}.$

Proof. By (32) and (35) we get
$\displaystyle\mathrm{CS}^{(\mathrm{F}_{1},\mathrm{F}_{2})}_{{\mathcal{H}}_{\text{chain}},{\mathscr{A}}^{\inf},{\mathscr{A}}^{\inf}}(f,\mu)\geqslant{\mathfrak{C}}_{\mu}^{(\mathrm{F}_{1},\mathrm{F}_{2})}(f),$
so we will prove the reverse inequality, that is,

$\displaystyle\displaystyle
L:=\sum_{i=1}^{l}\big{(}\mathrm{F}_{1}(\mathsf{A}^{\inf}(f|D_{i}),\mu(D_{i}))-\mathrm{F}_{2}(\mathsf{A}^{\inf}(f|D_{i-1}),\mu(D_{i}))\big{)}\leqslant{\mathfrak{C}}_{\mu}^{(\mathrm{F}_{1},\mathrm{F}_{2})}(f)$

for any $\displaystyle{\mathcal{D}}_{l}\in{\mathcal{H}}_{\text{chain}}$ with
$\displaystyle l\in[n].$ Since

$\displaystyle\sum_{i=1}^{l}\bigl{(}\mathrm{F}_{1}(a_{i},b_{i})-\mathrm{F}_{2}(a_{i-1},b_{i})\bigr{)}=\sum_{i=1}^{l-1}\bigl{(}\mathrm{F}_{1}(a_{i},b_{i})-\mathrm{F}_{2}(a_{i},b_{i+1})\bigr{)}+\mathrm{F}_{1}(a_{l},b_{l})-\mathrm{F}_{2}(a_{0},b_{1})$

for any nonnegative sequence $\displaystyle(a_{i})_{i=0}^{l}$ and
$\displaystyle(b_{i})_{i=1}^{l},$ we get $\displaystyle L=S_{1},$ where

$\displaystyle\displaystyle
S_{k}=\sum_{i=k}^{l-1}\big{(}\mathrm{F}_{1}(\mathsf{A}^{\inf}(f|D_{i}),\mu(D_{i}))-\mathrm{F}_{2}(\mathsf{A}^{\inf}(f|D_{i}),\mu(D_{i+1}))\big{)}+\mathrm{F}_{1}(\mathsf{A}^{\inf}(f|D_{l}),\mu(D_{l}))$

for $\displaystyle k\in[l],$ as $\displaystyle\mathrm{F}_{2}(0,\mu(D_{1}))=0.$
Let $\displaystyle k_{1}=\min\\{k\colon\mathsf{A}^{\inf}(f|D_{1})=f_{(k)}\\}.$
Clearly, $\displaystyle D_{1}\subseteq B_{(k_{1})}.$ Hence by monotonicity of
$\displaystyle\mathrm{F}_{1}$ we have

$\displaystyle\displaystyle L$
$\displaystyle\displaystyle\leqslant\mathrm{F}_{1}(f_{(k_{1})},\mu(B_{(k_{1})}))-\mathrm{F}_{2}(f_{(k_{1})},\mu(D_{2}))+S_{2}.$
(36)

Then there exists $\displaystyle k_{2}\in\\{k_{1}+1,\ldots,n\\}$ such that
$\displaystyle|D_{2}|=|B_{(k_{2})}|$ and
$\displaystyle\mathsf{A}^{\inf}(f|D_{2})\leqslant\mathsf{A}^{\inf}(f|B_{(k_{2})})$
(see the proof of Theorem 4.3). Thus

$\displaystyle\displaystyle L$
$\displaystyle\displaystyle\leqslant\mathrm{F}_{1}(f_{(k_{1})},\mu(B_{(k_{1})}))-\mathrm{F}_{2}(f_{(k_{1})},\mu(B_{(k_{2})}))+\mathrm{F}_{1}(\mathsf{A}^{\inf}(f|D_{2}),\mu(D_{2}))-\mathrm{F}_{2}(\mathsf{A}^{\inf}(f|D_{2}),\mu(D_{3}))+S_{3}.$

Since $\displaystyle\mathrm{F}_{1}$ and $\displaystyle\mathrm{F}_{2}$ are
pairwise $\displaystyle 2$-increasing,
$\displaystyle\mu(D_{3})\leqslant\mu(D_{2})$ and
$\displaystyle\mathsf{A}^{\inf}(f|B_{(k_{2})})=f_{(k_{2})},$ we get

$\displaystyle\displaystyle L$
$\displaystyle\displaystyle\leqslant\mathrm{F}_{1}(f_{(k_{1})},\mu(B_{(k_{1})}))-\mathrm{F}_{2}(f_{(k_{1})},\mu(B_{(k_{2})}))+\mathrm{F}_{1}(f_{(k_{2})},\mu(D_{2}))-\mathrm{F}_{2}(f_{(k_{2})},\mu(D_{3}))+S_{3}$
$\displaystyle\displaystyle=\mathrm{F}_{1}(f_{(k_{1})},\mu(B_{(k_{1})}))-\mathrm{F}_{2}(f_{(k_{1})},\mu(B_{(k_{2})}))+\mathrm{F}_{1}(f_{(k_{2})},\mu(B_{(k_{2})}))-\mathrm{F}_{2}(f_{(k_{2})},\mu(D_{3}))+S_{3}.$
(37)

Let $\displaystyle
M=\mathrm{F}_{1}(f_{(k_{1})},\mu(B_{(k_{1})}))-\mathrm{F}_{2}(f_{(k_{1})},\mu(B_{(k_{2})})).$
Then by $\displaystyle\mathrm{F}_{1}\geqslant\mathrm{F}_{2}$ and pairwise
$\displaystyle 2$-increasingness we get

$\displaystyle\displaystyle M$
$\displaystyle\displaystyle\leqslant\mathrm{F}_{1}(f_{(k_{1})},\mu(B_{(k_{1})}))-\mathrm{F}_{2}(f_{(k_{1})},\mu(B_{(k_{1}+1)}))+\mathrm{F}_{1}(f_{(k_{1})},\mu(B_{(k_{1}+1)}))-\mathrm{F}_{2}(f_{(k_{1})},\mu(B_{(k_{2})}))$
$\displaystyle\displaystyle\leqslant\mathrm{F}_{1}(f_{(k_{1})},\mu(B_{(k_{1})}))-\mathrm{F}_{2}(f_{(k_{1})},\mu(B_{(k_{1}+1)}))+\mathrm{F}_{1}(f_{(k_{1}+1)},\mu(B_{(k_{1}+1)}))-\mathrm{F}_{2}(f_{(k_{1}+1)},\mu(B_{(k_{2})}))$
$\displaystyle\displaystyle\leqslant\ldots\leqslant\sum_{i=k_{1}}^{k_{2}-1}\big{(}\mathrm{F}_{1}(f_{(i)},\mu(B_{(i)}))-\mathrm{F}_{2}(f_{(i)},\mu(B_{(i+1)}))\big{)}.$
(38)

Combining (4) and (4) we get

$\displaystyle\displaystyle L$
$\displaystyle\displaystyle\leqslant\sum_{i=k_{1}}^{k_{2}-1}\big{(}\mathrm{F}_{1}(f_{(i)},\mu(B_{(i)}))-\mathrm{F}_{2}(f_{(i)},\mu(B_{(i+1)}))\big{)}+\mathrm{F}_{1}(f_{(k_{2})},\mu(B_{(k_{2})}))-\mathrm{F}_{2}(f_{(k_{2})},\mu(D_{3}))+S_{3}.$

By repeating the same arguments, we obtain

$\displaystyle\displaystyle L$
$\displaystyle\displaystyle\leqslant\sum_{i=k_{1}}^{k_{l}-1}\big{(}\mathrm{F}_{1}(f_{(i)},\mu(B_{(i)}))-\mathrm{F}_{2}(f_{(i)},\mu(B_{(i+1)}))\big{)}+\mathrm{F}_{1}(f_{(k_{l})},\mu(B_{(k_{l})}))\leqslant{\mathfrak{C}}_{\mu}^{(\mathrm{F}_{1},\mathrm{F}_{2})}(f),$

where in the last inequality the formula (33) has been used. This completes
the proof. ∎

Let $\displaystyle\mathrm{C}$ be a copula. Similarly as in the proof of
Theorem 4.13, one can prove that the operator defined in (35) with
$\displaystyle\mathrm{F}_{1}|_{[0,1]^{2}}=\mathrm{F}_{2}|_{[0,1]^{2}}=\mathrm{C}$
coincides with
$\displaystyle{\mathfrak{C}}_{\mu}^{(\mathrm{C},\mathrm{C})}(f)$
($\displaystyle\mathrm{C}\mathrm{C}$-integral) for any $\displaystyle
f\in\mathbf{F}^{1}$ and any symmetric capacity $\displaystyle\mu.$

The operator
$\displaystyle{\mathfrak{C}}_{\mu}^{(\mathrm{F}_{1},\mathrm{F}_{2})}$ can also
be represented via $\displaystyle t$-level sets. Indeed, the equality

$\displaystyle\displaystyle{\mathfrak{C}}_{\mu}^{(\mathrm{F}_{1},\mathrm{F}_{2})}(f)=\sum_{i=1}^{n}\big{(}\mathrm{F}_{1}(f_{(i)},\mu(\\{f\geqslant
f_{(i)}\\}))-\mathrm{F}_{2}(f_{(i-1)},\mu(\\{f\geqslant f_{(i)}\\}))\big{)},$
(39)

where $\displaystyle(\cdot)\colon[n]\to[n]$ is a permutation such that
$\displaystyle 0=f_{(0)}\leqslant f_{(1)}\leqslant\ldots\leqslant f_{(n)},$
holds

  * •

for any $\displaystyle(f,\mu)\in\mathbf{F}\times\mathbf{M}$ whenever
$\displaystyle\mathrm{F}_{1}=\mathrm{F}_{2}$, or

  * •

for any $\displaystyle\mu\in\mathbf{M}$ and any
$\displaystyle\mathrm{F}_{1},\mathrm{F}_{2}$ such that
$\displaystyle\mathrm{F}_{1}\geqslant\mathrm{F}_{2}$ whenever $\displaystyle
f\in\mathbf{F}$ is decreasing or increasing.

Observe that the equality (39) need not hold for any $\displaystyle
f\in\mathbf{F}$ and any $\displaystyle\mathrm{F}_{1},\mathrm{F}_{2}$ such that
$\displaystyle\mathrm{F}_{1}\geqslant\mathrm{F}_{2}$ even for symmetric
$\displaystyle\mu\in\mathbf{M}.$

######  Example 4.14.

Let $\displaystyle X,$ $\displaystyle f$ and $\displaystyle\mu$ be the same as
in Example 4.6. Then

$\displaystyle\displaystyle\sum_{i=1}^{2}\big{(}\mathrm{F}_{1}(f_{(i)},\mu(\\{f\geqslant
f_{(i)}\\}))-\mathrm{F}_{2}(f_{(i-1)},\mu(\\{f\geqslant f_{(i)}\\}))\big{)}$
$\displaystyle\displaystyle=\mathrm{F}_{1}(0.5,2)-\mathrm{F}_{2}(0,2)+\mathrm{F}_{1}(0.5,2)-\mathrm{F}_{2}(0.5,2),$
$\displaystyle\displaystyle\sum_{i=1}^{2}\big{(}\mathrm{F}_{1}(f_{(i)},\mu(B_{(i)}))-\mathrm{F}_{2}(f_{(i-1)},\mu(B_{(i)}))\big{)}$
$\displaystyle\displaystyle=\mathrm{F}_{1}(0.5,2)-\mathrm{F}_{2}(0,2)+\mathrm{F}_{1}(0.5,1)-\mathrm{F}_{2}(0.5,1).$

The equality in (39) does not hold e.g., if
$\displaystyle\mathrm{F}_{1}=\mathrm{F}$ and
$\displaystyle\mathrm{F}_{2}=c\mathrm{F}$ for any $\displaystyle c\in(0,1)$
and any $\displaystyle\mathrm{F}\colon[0,\infty)^{2}\to[0,\infty)$ such that
$\displaystyle\mathrm{F}(0.5,2)\neq\mathrm{F}(0.5,1)$.

### Connection with $\displaystyle\mathfrak{IE}_{\mu}^{\circ,\mathsf{I}}$

Let
$\displaystyle\circ\colon[0,\infty)\times(-\infty,\infty)\to(-\infty,\infty).$
Define the following operator

$\displaystyle\displaystyle\mathfrak{IE}_{\mu}^{\circ,\mathsf{I}}(f)=\sum_{\emptyset\neq
D\subseteq X}\mathsf{I}(f,D)\circ\textsf{Mob}_{\mu}(D)$ (40)

for any $\displaystyle(f,\mu)\in\mathbf{F}\times\mathbf{M},$ where
$\displaystyle\mathsf{I}\colon\mathbf{F}\times\Sigma_{0}\to[0,\infty)$ is an
extended interaction operator satisfying the conditions:

  * $\displaystyle(I1)$

$\displaystyle\mathsf{I}(f,\\{i\\})=f(i)$ for any $\displaystyle
f\in\mathbf{F}$ and any $\displaystyle i\in[n]$;

  * $\displaystyle(I2)$

$\displaystyle\mathsf{I}(f,D)\leqslant\mathsf{I}(g,D)$ for any $\displaystyle
D\in\Sigma_{0}$ and any $\displaystyle f,g\in\mathbf{F}$ such that
$\displaystyle f(x)\leqslant g(x)$ for all $\displaystyle x\in D$;

  * $\displaystyle(I3)$

$\displaystyle\mathsf{I}(f,D)\geqslant\mathsf{I}(f,E)$ for any $\displaystyle
f\in\mathbf{F}$ and any $\displaystyle D,E\in\Sigma_{0}$ with $\displaystyle
D\subseteq E.$

Putting $\displaystyle\circ=\cdot$ and $\displaystyle\mathsf{I}$ being an
interaction operator ([17, Definition 5]) in (40), we get a discrete
inclusion-exclusion integral ([17, Theorem 3]). For a connection between
$\displaystyle\mathfrak{IE}^{\circ,\mathsf{I}}_{\mu}$ and the Choquet-Sugeno-
like operator of the form (14) we first explain a relationship between the
extended interaction operator and conditional aggregation operator.

######  Example 4.15.

  * (i)

The mapping $\displaystyle\mathsf{A}(f|D)=(\textstyle{\sum_{i\in
D}f(i)^{p}})^{1/p}$ with $\displaystyle p\in[1,\infty)$ is the conditional
aggregation operator which is not interaction operator since it violates
$\displaystyle(I3).$

  * (ii)

The Łukasiewicz t-conorm
$\displaystyle\mathsf{A}(f|D)=1\wedge\textstyle{\sum_{i\in D}f(i)}$ is the
conditional aggregation operator violating the properties $\displaystyle(I1)$
and $\displaystyle(I3).$

These examples with the next proposition claim that the concept of conditional
aggregation is more general than that of extended interaction operators. For
that purpose we introduce the following notion: a conditional aggregation
operator $\displaystyle\mathsf{A}(\cdot|\cdot)$ is called conjunctive if
$\displaystyle\mathsf{A}(f|D)\leqslant\mathsf{A}^{\inf}(f|D)$ for any
$\displaystyle D\in\Sigma_{0}$ and any $\displaystyle f\in\mathbf{F}.$

######  Proposition 4.16.

Let $\displaystyle D\in\Sigma_{0}.$ Then
$\displaystyle\mathsf{I}(f,D)=\mathsf{A}(f|D)$ for any $\displaystyle
f\in\mathbf{F},$ where $\displaystyle\mathsf{A}$ is a conjunctive conditional
aggregation operator.

Proof. In order to prove that an extended interaction operator
$\displaystyle\mathsf{I}$ is a conditional aggregation operator, only
condition $\displaystyle(C2)$ has to be verified. Let $\displaystyle
D\in\Sigma_{0}.$ Clearly, $\displaystyle D$ can be written as a finite union
of singletons, let us say $\displaystyle D=\textstyle{\bigcup_{j\in
D}\\{j\\}}.$ Then for any $\displaystyle f\in\mathbf{F}$ from
$\displaystyle(I3)$ and $\displaystyle(I1)$ we have
$\displaystyle\mathsf{I}(\mathbbm{1}_{D^{c}},D)\leqslant\mathsf{I}(\mathbbm{1}_{D^{c}},\\{j\\})=\mathbbm{1}_{D^{c}}(j)=0$
for each $\displaystyle j\in D.$ This yields that $\displaystyle\mathsf{I}$
satisfies $\displaystyle(C2).$ To prove conjunctivity, for $\displaystyle
D\in\Sigma_{0}$ put $\displaystyle f(j)=\textstyle{\min_{i\in D}f(i)}.$ Then
$\displaystyle j\in D,$ and

$\displaystyle\mathsf{I}(f,D)\leqslant\mathsf{I}(f,\\{j\\})=f(j)=\mathsf{A}^{\inf}(f|D).$

This completes the proof. ∎

Using Proposition 4.16, the connection between
$\displaystyle\mathfrak{IE}^{\circ,\mathsf{I}}_{\mu}$ and (14) is immediate.

######  Theorem 4.17.

Let
$\displaystyle\circ\colon[0,\infty)\times(-\infty,\infty)\to(-\infty,\infty)$
and $\displaystyle{\mathscr{A}}$ be a FCA consisting of all conjunctive
conditional aggregation operators. Then the operator
$\displaystyle\mathfrak{IE}^{\circ,\mathsf{I}}_{\mu}(f)$ with
$\displaystyle\mathsf{I}\in{\mathscr{A}}$ coincides with the Choquet-Sugeno-
like operator (14) for any
$\displaystyle(f,\mu)\in\mathbf{F}\times\mathbf{M}.$

##  5 Selected properties of the Choquet-Sugeno-like operator

In this section several properties of the Choquet-Sugeno-like operator will be
examined with respect to function $\displaystyle\mathrm{L}.$ Let us summarize
the functions $\displaystyle\mathrm{L}=\mathrm{L}(x,y,z,w)$ that have been
used in the paper:

$\displaystyle\displaystyle 1.\quad$
$\displaystyle\displaystyle\mathrm{L}_{1}(x,y,z,w)=|x-y|^{p}\quad$ (Example
3.9); $\displaystyle\displaystyle 2.\quad$
$\displaystyle\displaystyle\mathrm{L}_{2}(x,y,z,w)=x\circ z\quad$ (Examples
3.4–3.6 with $\displaystyle\mathrm{F}=\circ$); $\displaystyle\displaystyle
3.\quad$ $\displaystyle\displaystyle\mathrm{L}_{3}(x,y,z,w)=y\circ w\quad$
(formula (14)); $\displaystyle\displaystyle 4.\quad$
$\displaystyle\displaystyle\mathrm{L}_{4}(x,y,z,w)=x\circ(z-w)_{+}\quad$
$\displaystyle\displaystyle\text{(formulas \eqref{n:c1b} and \eqref{n:c5a})};$
$\displaystyle\displaystyle 5.\quad$
$\displaystyle\displaystyle\mathrm{L}_{5}(x,y,z,w)=(x-y)_{+}\circ z\quad$
(formula (15)); $\displaystyle\displaystyle 6.\quad$
$\displaystyle\displaystyle\mathrm{L}_{6}(x,y,z,w)=\delta(x,y)\circ z\quad$
$\displaystyle\displaystyle\text{(formula \eqref{n:d1b})};$
$\displaystyle\displaystyle 7.\quad$
$\displaystyle\displaystyle\mathrm{L}_{7}(x,y,z,w)=\mathrm{F}_{1}(x,z)-\mathrm{F}_{2}(y,z)\quad$
$\displaystyle\displaystyle\text{(formula \eqref{n:cf3})},$

where $\displaystyle\delta,\circ,\mathrm{F}_{1},\mathrm{F}_{2}$ have been
defined at the indicated places of the article. To simplify the notation, we
will write just $\displaystyle\mathrm{CS}^{\mathrm{L}}(f)$ instead of
$\displaystyle\mathrm{CS}^{\mathrm{L}}_{{\mathcal{H}},{\mathscr{A}},\widehat{{\mathscr{A}}}}(f,\mu,\widehat{\mu}).$
Proofs of the results presented in this section are immediate when
representing the Choquet-Sugeno-like operator (5) in the following way

$\displaystyle\displaystyle\mathrm{CS}^{\mathrm{L}}(f)$
$\displaystyle\displaystyle=\sup_{{\mathcal{D}}\in{\mathcal{H}}}\Big{\\{}\sum_{(C,D)\in{\mathcal{R}},\,C,D\neq\emptyset}\mathrm{L}\big{(}\mathsf{A}(f|C),\widehat{\mathsf{A}}(f|D),\mu(C),\widehat{\mu}(D)\big{)}+\sum_{(C,\emptyset)\in{\mathcal{R}},\,C\neq\emptyset}\mathrm{L}\big{(}\mathsf{A}(f|C),0,\mu(C),0\big{)}$
$\displaystyle\displaystyle\qquad\qquad+\sum_{(\emptyset,D)\in{\mathcal{R}},\,D\neq\emptyset}\mathrm{L}\big{(}0,\widehat{\mathsf{A}}(f|D),0,\widehat{\mu}(D)\big{)}+\sum_{(\emptyset,\emptyset)\in{\mathcal{R}}}\mathrm{L}(0,0,0,0)\Big{\\}},$
(41)

as
$\displaystyle\mathsf{A}(\cdot|\emptyset)=\widehat{\mathsf{A}}(\cdot|\emptyset)=0$
and $\displaystyle\widehat{\mu}(\emptyset)=0.$

######  Proposition 5.1.

If $\displaystyle\mathrm{L}(0,0,z,w)=0$ for all $\displaystyle z,w,$ then
$\displaystyle\mathrm{CS}^{\mathrm{L}}(0_{X})=0.$

Proof. The statement follows from (5) and the fact that
$\displaystyle\mathsf{A}(0_{X}|D)=0=\widehat{\mathsf{A}}(0_{X}|D)$ for any
$\displaystyle D\in\Sigma_{0},$ where
$\displaystyle\mathsf{A}\in{\mathscr{A}}$ and
$\displaystyle\widehat{\mathsf{A}}\in\widehat{{\mathscr{A}}}$ (see [2,
Proposition 3.3 (b)]). ∎

######  Example 5.2.

The assumption in Proposition 5.1 is satisfied for the following functions:

  * •

$\displaystyle\mathrm{L}_{1}$;

  * •

$\displaystyle\mathrm{L}_{2}-\mathrm{L}_{6}$ whenever $\displaystyle 0\circ
b=0$ for any $\displaystyle b$;

  * •

$\displaystyle\mathrm{L}_{7}$ whenever
$\displaystyle\mathrm{F}_{1}(0,b)=\mathrm{F}_{2}(0,b).$

######  Proposition 5.3.

(Monotonicity) Assume that $\displaystyle\mathrm{L}(x,y,z,w)$ is nondecreasing
in $\displaystyle x$ and $\displaystyle y$ for any fixed $\displaystyle z,w.$
Then
$\displaystyle\mathrm{CS}^{\mathrm{L}}(f)\leqslant\mathrm{CS}^{\mathrm{L}}(g)$
whenever $\displaystyle f\leqslant g.$

Proof. Use the equality (5) and the property $\displaystyle(C1)$ (see
Definition 2.1). ∎

######  Example 5.4.

The assumption of Proposition 5.3 is satisfied for functions
$\displaystyle\mathrm{L}_{2}-\mathrm{L}_{4}$ whenever $\displaystyle x\mapsto
x\circ b$ is nondecreasing for any $\displaystyle b.$

We say that a conditional aggregation operator
$\displaystyle\mathsf{A}(\cdot|D)$ is homogeneous if
$\displaystyle\mathsf{A}(\alpha f|D)=\alpha\mathsf{A}(f|D)$ for any
$\displaystyle\alpha\in[0,\infty)$ and $\displaystyle f\in\mathbf{F}.$ A
family $\displaystyle{\mathscr{A}}$ of conditional aggregation operators is
said to be homogeneous if $\displaystyle\mathsf{A}(\cdot|D)\in{\mathscr{A}}$
is homogeneous for any $\displaystyle D\in\Sigma_{0}.$

######  Proposition 5.5.

(Homogeneity) Let $\displaystyle{\mathscr{A}}$ and
$\displaystyle\widehat{{\mathscr{A}}}$ be homogeneous FCAs. If
$\displaystyle\mathrm{L}(\alpha x,\alpha y,z,w)=\alpha\mathrm{L}(x,y,z,w)$ for
all $\displaystyle\alpha\in[0,\infty)$ and all $\displaystyle x,y,z,w,$ then
$\displaystyle f\mapsto\mathrm{CS}^{\mathrm{L}}(f)$ is a homogeneous operator.

######  Example 5.6.

The assumption is satisfied for:

  * •

$\displaystyle\mathrm{L}_{1}$ for $\displaystyle p=1$;

  * •

$\displaystyle\mathrm{L}_{2}-\mathrm{L}_{5}$ if $\displaystyle(\alpha a)\circ
b=\alpha(a\circ b)$ for any $\displaystyle\alpha,a,b$;

  * •

$\displaystyle\mathrm{L}_{6}$ if $\displaystyle\delta(\alpha x,\alpha y)\circ
z=\alpha(\delta(x,y)\circ z)$ for any $\displaystyle\alpha,x,y,z$;

  * •

$\displaystyle\mathrm{L}_{7}$ if $\displaystyle x\mapsto\mathrm{F}_{1}(x,b)$
and $\displaystyle x\mapsto\mathrm{F}_{2}(x,b)$ are homogeneous for any
$\displaystyle b.$

We say that a conditional aggregation operator
$\displaystyle\mathsf{A}(\cdot|D)$ is subadditive if
$\displaystyle\mathsf{A}(f+g|D)\leqslant\mathsf{A}(f|D)+\mathsf{A}(g|D)$ for
any $\displaystyle f,g\in\mathbf{F}.$ A FCA $\displaystyle{\mathscr{A}}$ is
said to be subadditive if $\displaystyle\mathsf{A}(\cdot|D)\in{\mathscr{A}}$
is subadditive for any $\displaystyle D\in\Sigma_{0}.$

######  Proposition 5.7.

(Subadditivity) Suppose that $\displaystyle{\mathscr{A}}$ and
$\displaystyle\widehat{{\mathscr{A}}}$ are subadditive FCAs. If
$\displaystyle\mathrm{L}(x+a,y+b,z,w)\leqslant\mathrm{L}(x,y,z,w)+\mathrm{L}(a,b,z,w)$
for all $\displaystyle x,y,z,w,a,b,$ then the mapping $\displaystyle
f\mapsto\mathrm{CS}^{\mathrm{L}}(f)$ is subadditive.

######  Example 5.8.

The following functions $\displaystyle\mathrm{L}$ are subadditive for the
first and second coordinate:

  * •

$\displaystyle\mathrm{L}_{1}$ with $\displaystyle p=1$;

  * •

$\displaystyle\mathrm{L}_{2}-\mathrm{L}_{4}$ are subadditive if $\displaystyle
x\mapsto x\circ b$ is subadditive for any $\displaystyle b$;

  * •

$\displaystyle\mathrm{L}_{5}$ if $\displaystyle x\mapsto x\circ z$ is
nondecreasing and subadditive for any $\displaystyle z$;

  * •

$\displaystyle\mathrm{L}_{6}$ if $\displaystyle\delta(x+a,y+b)\circ
z\leqslant\delta(x,y)\circ z+\delta(a,b)\circ z$ for any $\displaystyle
x,y,z,a,b.$

######  Proposition 5.9.

(Convexity) Let $\displaystyle f\mapsto\mathsf{A}(f|D)$ and $\displaystyle
f\mapsto\widehat{\mathsf{A}}(f|D)$ be convex for any $\displaystyle
D\in\Sigma_{0},$ where $\displaystyle\mathsf{A}\in{\mathscr{A}}$ and
$\displaystyle\widehat{\mathsf{A}}\in\widehat{{\mathscr{A}}}.$ If
$\displaystyle\mathrm{L}(\lambda x+(1-\lambda)a,\lambda
y+(1-\lambda)b,z,w)\leqslant\lambda\mathrm{L}(x,y,z,w)+(1-\lambda)\mathrm{L}(a,b,z,w)$
for all $\displaystyle\lambda\in(0,1)$ and all $\displaystyle x,y,z,w,a,b,$
then $\displaystyle f\mapsto\mathrm{CS}^{\mathrm{L}}(f)$ is convex.

######  Example 5.10.

The assumption in Proposition 5.9 is satisfied for

  * •

$\displaystyle\mathrm{L}_{1}$ if $\displaystyle p\geqslant 1$;

  * •

$\displaystyle\mathrm{L}_{2}-\mathrm{L}_{4}$ whenever $\displaystyle x\mapsto
x\circ b$ is convex for any $\displaystyle b$;

  * •

$\displaystyle\mathrm{L}_{5}$ whenever $\displaystyle x\mapsto x\circ z$ is
nondecreasing and convex for any $\displaystyle z.$

We say that a FCA $\displaystyle{\mathscr{A}}$ is idempotent if
$\displaystyle\mathsf{A}(b\mathbbm{1}_{X}|D)=b$ for any $\displaystyle
b\in[0,\infty)$ and any $\displaystyle\mathsf{A}(\cdot|D)\in{\mathscr{A}}$
with $\displaystyle D\in\Sigma_{0}$ (see [2, Proposition 3.10]). Obviously,
$\displaystyle{\mathscr{A}}^{\inf}$ and $\displaystyle{\mathscr{A}}^{\sup}$
are idempotent FCAs.

######  Proposition 5.11.

(Idempotency) Let $\displaystyle{\mathscr{A}}$ and
$\displaystyle\widehat{{\mathscr{A}}}$ be idempotent FCAs. Then
$\displaystyle\mathrm{CS}^{\mathrm{L}}(b\mathbbm{1}_{X})=b$ for all
$\displaystyle b\in[0,\infty)$ if and only if

$\displaystyle\displaystyle\sup_{{\mathcal{D}}\in{\mathcal{H}}}\Big{\\{}$
$\displaystyle\displaystyle\sum_{(C,D)\in{\mathcal{R}},\,D\neq\emptyset}\mathrm{L}(b,b,\mu(C),\widehat{\mu}(D))+\sum_{(C,\emptyset)\in{\mathcal{R}},\,C\neq\emptyset}\mathrm{L}(b,0,\mu(C),0)$
$\displaystyle\displaystyle+\sum_{(\emptyset,D)\in{\mathcal{R}},\,D\neq\emptyset}\mathrm{L}(0,b,0,\widehat{\mu}(D))+\sum_{(\emptyset,\emptyset)\in{\mathcal{R}}}\mathrm{L}(0,0,0,0)\Big{\\}}=b$
(42)

for any $\displaystyle b\in[0,\infty).$

For the special choice of L’s the condition (5.11) can be simplified as the
following examples demonstrate.

  * •

Consider $\displaystyle\mathrm{L}_{1}.$ If $\displaystyle{\mathscr{A}}$ and
$\displaystyle\widehat{{\mathscr{A}}}$ are idempotent FCAs, then the
$\displaystyle p$-variation of any constant function is equal to 0 (see
Example 3.9), so the $\displaystyle p$-variation is not idempotent, as
expected.

  * •

Consider $\displaystyle\mathrm{L}_{2}$ such that $\displaystyle x\circ
0=0=0\circ z$ for any $\displaystyle x,z.$ Then (5.11) takes the form

$\displaystyle\displaystyle\sup_{{\mathcal{D}}\in{\mathcal{H}}}\sum_{(C,D)\in{\mathcal{R}},\,C\neq\emptyset}b\circ\mu(C)=b.$
(43)

If $\displaystyle{\mathcal{R}}=\\{(D,D)\colon D\in{\mathcal{D}}\\}$ for
$\displaystyle{\mathcal{D}}\in{\mathcal{H}}=\\{\\{D\\}\colon
D\in\Sigma_{0}\\}$ (see Example 3.4), then (43) can be rewritten as follows
$\displaystyle\textstyle{\sup_{C\in\Sigma_{0}}\\{b\circ\mu(C)\\}}=b$ for any
$\displaystyle b.$ For $\displaystyle\mathrm{L}_{3}$ we can apply a similar
approach.

  * •

Consider $\displaystyle\mathrm{L}_{4},$
$\displaystyle{\mathcal{R}}={\mathcal{R}}^{+}$ and
$\displaystyle{\mathcal{H}}={\mathcal{H}}_{\text{chain}}$ (cf. section devoted
to $\displaystyle{\mathfrak{C}}^{\mu,\widehat{\mu}}_{\circ}$). Then the
condition (5.11) has the following form

$\displaystyle\displaystyle\sup_{{\mathcal{D}}\in{\mathcal{H}}_{\text{chain}}}\sum_{i=1}^{l}b\circ(\mu(D_{i})-\widehat{\mu}(D_{i+1}))=b$
(44)

for any $\displaystyle b.$ The condition (44) is satisfied if
$\displaystyle\circ=\cdot$ and
$\displaystyle\mu=\widehat{\mu}\in\mathbf{M}^{1}$ (cf. [18, Proposition 14]).

  * •

Consider $\displaystyle\mathrm{L}_{5},$ $\displaystyle 0\circ z=0$ for any
$\displaystyle z$ and $\displaystyle{\mathcal{R}}=\\{(D,D^{c})\colon
D\in{\mathcal{D}}\\}$ for
$\displaystyle{\mathcal{D}}\in{\mathcal{H}}_{\text{one}}$ (cf. Example 3.8).
Then (5.11) reduces to the equality $\displaystyle b\circ\mu(X)=b$ for any
$\displaystyle b\in[0,\infty).$

  * •

Consider $\displaystyle\mathrm{L}_{6},$ $\displaystyle 0\circ z=0$ for any
$\displaystyle z,$ $\displaystyle{\mathcal{H}}={\mathcal{H}}_{\text{chain}}$
and $\displaystyle{\mathcal{R}}={\mathcal{R}}^{-}$ (cf. section devoted to
$\displaystyle{\mathfrak{C}}^{\circ,\delta}_{\mu}$). Then the condition (5.11)
takes the form

$\displaystyle\displaystyle\sup_{D\in\Sigma_{0}}\\{\delta(b,0)\circ\mu(D)\\}=b$
(45)

for any $\displaystyle b.$ The condition (45) is true for nondecreasing
$\displaystyle x\mapsto a\circ x$ for any $\displaystyle a$ such that
$\displaystyle\delta(a,0)\circ\mu(X)=a$ for any $\displaystyle a$ (cf. [8,
Theorem 3.31]).

  * •

Consider $\displaystyle\mathrm{L}_{7},$
$\displaystyle{\mathcal{H}}={\mathcal{H}}_{\text{chain}}$ and
$\displaystyle{\mathcal{R}}={\mathcal{R}}^{-}$ (cf. section devoted to
$\displaystyle{\mathfrak{C}}^{(\mathrm{F}_{1},\mathrm{F}_{2})}_{\mu}$). Then
the formula (5.11) takes the form

$\displaystyle\displaystyle\sup_{{\mathcal{D}}\in{\mathcal{H}}_{\text{chain}}}\sum_{i=1}^{l}\bigl{(}\mathrm{F}_{1}(b,\mu(D_{i}))-\mathrm{F}_{2}(b,\mu(D_{i}))\bigr{)}=b$

for any $\displaystyle b\in[0,\infty).$

##  6 Conclusion

In this paper we have indicated a way how to look at different operators with
respect to a nonadditive measure from a new (in some sense unified)
perspective. We have defined an operator generalizing Sugeno-like operator
(Example 3.4), generalized Lebesgue integral (Example 3.5),
$\displaystyle\mathrm{F}$-decomposition integral with respect to a finite
partition decomposition system (Example 3.6), Lovász extension (Example 3.7)
and generalized $\displaystyle p$-variation (Example 3.9). Moreover, we have
given relationships with several functionals generalizing the discrete Choquet
integral expressions

$\displaystyle\displaystyle(C)\int\limits_{X}f\,{\mathrm{d}}\mu$
$\displaystyle\displaystyle=\sum_{i=1}^{n}f_{(i)}(\mu(B_{(i)})-\mu(B_{(i+1)}))\qquad$
$\displaystyle\displaystyle\text{(cf.
$\displaystyle{\mathfrak{C}}_{\circ}^{\mu,\widehat{\mu}}(f)$)},$
$\displaystyle\displaystyle=\sum_{i=1}^{n}(f_{(i)}-f_{(i-1)})\mu(B_{(i)})\quad$
$\displaystyle\displaystyle\text{(cf.
$\displaystyle{\mathfrak{C}}_{\mu}^{\circ,\delta}(f)$)},$
$\displaystyle\displaystyle=\sum_{i=1}^{n}\bigl{(}f_{(i)}\mu(B_{(i)})-f_{(i-1)}\mu(B_{(i)})\bigr{)}\qquad\qquad$
$\displaystyle\displaystyle\text{(cf.
$\displaystyle{\mathfrak{C}}_{\mu}^{(\mathrm{F}_{1},\mathrm{F}_{2})}(f)$)}.$

All of this has been possible thanks to the conditional aggregation operator
defined in [2] and the dependence relation between conditional sets proposed
in the present paper. The relation can generate different aggregation styles
in decision making and can be used in as diverse areas as graph theory, neural
networks and fuzzy theory. Thus, describing new relations between conditional
sets may produce new operators interesting both from a theoretical point of
view as well as for applications.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

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