1. Introduction
The concepts of overlap functions and grouping functions were firstly introduced by Bustince et al. in [1] [2] and [3], respectively. Overlap functions and grouping functions are two particular cases of bivariate continuous aggregation functions [4] [5]. Those two concepts have been applied to some interesting problems, for example, image processing [1] [6], classification [7] [8] and decision making [3] [9]. In recent years, some extended forms of overlap functions and grouping functions were presented, for example, n-Dimensional overlap functions and grouping functions [10], general overlap functions [11]. Overlap functions and grouping functions can be constructed by using additive generator pairs [12] or multiplicative generator pairs [13]. Xie [14] proposed the concepts of multiplicative generator pairs of n-dimensional overlap functions and presented the condition under which the multiplicative generator pairs can generate an n-dimensional overlap function. In [10], some open problems were presented. One of the open problems is “to search the construction of n-dimensional overlapping functions based on bi-dimensional overlapping functions”. So far, this open problem has not been solved. In this paper, we try to solve this open problem. One characteristic of the conjunction operator
satisfies associativity and commutativity. We construct n-dimensional overlap functions
by means of
, and study their basic properties.
The rest of this paper is organized as follows. In Section 2, we review some concepts and results about overlap functions and n-dimensional overlap functions, which will be used throughout this paper. In Section 3, we mainly introduce some new results on overlap functions and n-dimensional overlap functions. In Section 4, one construction method of n-dimensional overlap functions based on bivariate overlap functions is discussed. We provide some conclusions in Section 5.
2. Preliminaries
In this section, we recall some concepts and properties of bivariate overlap functions and n-dimensional overlap functions which shall be needed in the sequel.
Definition 2.1 (See Bustince et al. [1] ). A bivariate function
is said to be an overlap function if it satisfies the following conditions:
(O1)
is commutative;
(O2)
iff
;
(O3)
iff
;
(O4)
is increasing;
(O5)
is continuous.
Example 2.1 (See Qiao and Hu [15] ). For any
, consider the bivariate function
given by
for all
. Then it is an overlap function and we call it
-product overlap function, here. It is obvious that 1-product overlap function is the product t-norm. Moreover, for any
, the
-product overlap function is neither associative nor has 1 as neutral element. Therefore, it is not a t-norm.
Definition 2.2. (See Dimuro and Bedregal [16] ) An overlap function
satisfies the Property 1-section deflation if
(O6)
, and the Property 1-section inflation if
(O7)
.
An overlap function
satisfies the Property diagonal inflation [17] if
(O8)
for all
.
Denote by
the set of all overlap functions. Then
with the ordering
defined for
by
if and only if
for all
, is a lattice [16].
Lemma 2.1 (See Wang and Liu [18] ). Let
be an overlap function, and
be a strictly increasing automorphism. Then
is an overlap function given by
for all
.
Definition 2.3 (See Bustince et al. [2] ). Let
be a mapping and
.
is homogeneous of order
if for any
and for any
such that
the identity
holds.
An n-ary aggregation function
is said to be idempotent if
for any
.
Definition 2.4 (See Dimuro and Bedregal [19] ). An overlap function
is said to be Archimedean if, for each
, there exists
such that
, where
is
and
.
Lemma 2.2 (See Dimuro and Bedregal [19] ). Let
be an Archimedean overlap function. Then, for all
, it holds that
.
Definition 2.5 (See Gómez et al. [10] ). An n-dimensional aggregation function
is an n-dimensional overlap function if and only if:
.
is symmetric.
.
if and only if
.
.
if and only if
for all
.
.
is increasing.
.
is continuous.
Let us denote by
the set of all n-dimensional overlap functions. The set
is a lattice with the ordering
defined for
as
if and only if
for all
[10].
Lemma 2.3 (See Gómez et al. [10] ). Let
be an automorphism. Then, for every overlap function
,
and
are also overlap functions.
In this paper, the overlap function
will be denoted by
, i.e.,
.
Definition 2.6 (See Gómez et al. [10] ). Let
be a mapping and let
be a positive value. Then, the function
is homogeneous of order
if and only if for any
and for any
(with
for all
) the identity
holds.
3. Some New Results on Overlap Functions and n-Dimensional Overlap Functions
In this section, we mainly present some new results on overlap functions and n-dimensional overlap functions. These new results mainly reflect three properties: 1-section deflation, 1-section inflation and diagonal inflation on overlap functions and n-dimensional overlap functions.
Proposition 3.1. Let
be two overlap functions and
. If
satisfies the Property 1-section deflation, then
also satisfies the Property 1-section deflation.
Proof. Since
, if
satisfies the Property 1-section deflation, then for any
, one has that
. Hence
satisfies the Property 1-section deflation.
Proposition 3.2. Let
be two overlap functions and
. If
satisfies the Property 1-section inflation (or diagonal inflation), then
also satisfies the Property 1-section inflation (or diagonal inflation).
Proof. It can be proven in a similar way as that of Proposition 3.1.
Proposition 3.3. Let
be an overlap function. If
satisfies the Property 1-section deflation (1-section inflation or diagonal inflation), then
also satisfies the Property 1-section deflation (1-section inflation or diagonal inflation).
Proof. We only verify that the Property 1-section deflation. The other two properties can be verified in a similar way.
If
satisfies the Property 1-section deflation, then for any
,
.
Hence
satisfies the Property 1-section deflation.
Now, we extend three properties 1-section deflation, 1-section inflation and diagonal inflation to the n-dimensional case (
).
Definition 3.1. An n-dimensional overlap function
satisfies the Property 1-section deflation if
(
)
, and the Property 1-section inflation if
(
)
, and the Property diagonal inflation if
(
)
.
One can extend
in Lemma 2.1 to the n-dimensional case
.
Proposition 3.4 Let
be an n-dimensional overlap function, and
be a strictly increasing automorphism. Then
is an n-dimensional overlap function given by
for all
.
With similar Propositions 3.1 - 3.3, we easy to get the following Propositions.
Proposition 3.5. Let
be two n-dimensional overlap functions and
. If
satisfies the Property 1-section deflation, then
also satisfies the Property 1-section deflation.
Proposition 3.6. Let
be two n-dimensional overlap functions and
. If
satisfies the Property 1-section inflation (or diagonal inflation), then
also satisfies the Property 1-section inflation (or diagonal inflation).
Proposition 3.7. Let
be an n-dimensional overlap function. If
satisfies the Property 1-section deflation (1-section inflation or diagonal inflation), then
also satisfies the Property 1-section deflation (1-section inflation or diagonal inflation).
4. Constructing n-Dimensional Overlap Functions Based on Bivariate Overlap Functions
In this section, we mainly introduce the construction method of n-dimensional overlap functions based on bivariate overlap functions.
Proposition 4.1. Let
be a bivariate overlap function. Then the function
defined as
is an n-dimensional overlap function.
Proof.
. It is obviously that
is symmetric, because
is symmetric.
.
.
.
for all
,
for all
.
and
obviously hold.
Example 4.1. By use of
in Example 2.1, we can construct an 3-dimensional overlap function
as follows
Proposition 4.2. Let
be the idempotent element of bivariate overlap function
. Then
is also the idempotent element of
.
Proof. Let
be the idempotent element of
, then
.
Hence
is the idempotent element of
.
Proposition 4.3. Let
be an Archimedean overlap function. Then, for all
, it holds that
.
Proof. Let
be an Archimedean overlap function, by Lemma 2.2, for all
, we have
.
Proposition 4.4. Let
be a bivariate overlap function and
be an automorphism. Then
(1)
Proof.
Proposition 4.5. Let
be a bivariate overlap function and
is homogeneous of order
. Then
is also homogeneous of order
.
Proof. For any
and for any
Proposition 4.6. Let
be a bivariate overlap function. If
satisfies the Property 1-section deflation, then
also satisfies the Property 1-section deflation.
Proof. If
satisfies the Property 1-section deflation, then for any
, we have that
Therefor,
satisfies the Property 1-section deflation.
Similar to Proposition 4.6, we can get the following proposition.
Proposition 4.7. Let
be a bivariate overlap function. If
satisfies the Property 1-section inflation (or diagonal inflation), then
also satisfies the Property 1-section inflation (or diagonal inflation).
5. Conclusion
In this paper, we first introduce some new results on 1-section deflation, 1-section inflation and diagonal inflation. Next, three properties 1-section deflation, 1-section inflation and diagonal inflation are extended to the n-dimensional case (
), and the corresponding results are presented. Finally, we focus on one construction method of n-dimensional overlap functions
based on bivariate overlap functions and discuss their main properties, and well solve the open problem “to search the construction of n-dimensional overlapping functions based on bi-dimensional overlapping functions” in [10]. Because of the duality of n-dimensional overlap and grouping functions, one can also construct n-dimensional grouping functions based on bivariate grouping functions in a similar way.
Acknowledgements
This research was supported by National Nature Science Foundation of China (Grant Nos. 61763008, 11661028, 11661030).