=Paper=
{{Paper
|id=Vol-3074/paper01
|storemode=property
|title=General admissibly ordered interval-valued overlap functions
|pdfUrl=https://ceur-ws.org/Vol-3074/paper01.pdf
|volume=Vol-3074
|authors=Tiago Asmus, Graçaliz Dimuro, José Antonio Sanz Delgado, Jonata Wieczynski, Giancarlo Lucca,Humberto Bustince
|dblpUrl=https://dblp.org/rec/conf/wilf/AsmusDDWLB21
}}
==General admissibly ordered interval-valued overlap functions==
https://ceur-ws.org/Vol-3074/paper01.pdf
General Admissibly Ordered Interval-valued Overlap
Functions
Tiago da Cruz Asmus1,2 , Graçaliz Pereira Dimuro1,3 , José Antonio Sanz1 ,
Jonata Wieczynski1,4 , Giancarlo Lucca5 and Humberto Bustince1
1
Departamento de Estadística, Informática y Matemáticas, Universidad Publica de Navarra, Pamplona, Spain
2
Instituto de Matemática, Estatística e Física, Universidade Federal do Rio Grande, Rio Grande, Brazil
3
Centro de Ciências Computacionais, Universidade Federal do Rio Grande, Rio Grande, Brazil
4
Programa de Pós-Graduação em Computação, Universidade Federal do Rio Grande, Rio Grande, Brazil
5
Programa de Pós-Graduação em Modelagem Computacional, Universidade Federal do Rio Grande, Rio Grande, Brazil
Abstract
Overlap functions are a class of aggregation functions that measure the overlapping degree between
two values. They have been successfully applied in several problems in which associativity is not
required, such as classification and image processing. Some generalizations of overlap functions were
proposed for applications in problems with more than two classes, such as 𝑛-dimensional and general
overlap functions. To measure the overlapping of interval data, interval-valued overlap functions were
defined, and, later, they were also generalized in the form of 𝑛-dimensional and general iv-overlap
functions. In order to apply some of those concepts in problems with interval data considering the
use of admissible orders, which are total orders that refine the most used partial order for intervals,
𝑛-dimensional admissibly ordered iv-overlap functions were recently introduced, proving to be suitable
to be applied in classification problems. However, the sole construction method presented for this kind
of function do not allow the use of the well known lexicographical orders. So, in this work we combine
previous developments to introduce general admissibly ordered iv-overlap functions, present different
construction methods for them and how to combine such methods, showcasing the flexibility of this
approach, while also being compatible with the lexicographical orders.
1. Introduction
Overlap functions are aggregation functions, introduced in the context of image processing
problems, to measure the overlapping between classes [1]. They have been studied in the
literature by many authors, mainly because of either the advantages they present over t-norms
[2, 3] or their great applicability, as in: fuzzy rule-based classification [4, 5] and decision making
[6].
WILF 2021: The 13th International Workshop on Fuzzy Logic and Applications, December 20–22, 2021, Vietri sul Mare,
Italy
" tiago.dacruz@unavarra.es (T. d. C. Asmus); gracalizdimuro@furg.br (G. P. Dimuro);
joseantonio.sanz@unavarra.es (J. A. Sanz); jonatacw@gmail.com (J. Wieczynski); giancarlolucca@furg.br
(G. Lucca); bustince@unavarra.es (H. Bustince)
0000-0002-7066-7156 (T. d. C. Asmus); 0000-0001-6986-9888 (G. P. Dimuro); 0000-0002-1427-9909 (J. A. Sanz);
0000-0002-1279-6195 (H. Bustince)
© 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org)
The concept of 𝑛-dimensional overlap functions was introduced [7] to allow the application
of overlap functions in problems with multiple classes. By relaxing the boundary conditions,
general overlap functions were defined, showing good behaviour in classification problems [8].
When working with fuzzy systems, one may face the problem regarding the uncertainty in
assigning the values of the membership degrees or defining the membership functions that are
adopted in the system. In the literature, a proposed solution is given by the use of interval-valued
(iv) fuzzy sets (IVFSs) [9], where membership degrees are represented by intervals, whose widths
express such uncertainty [10]. IVFSs have been successfully applied in different fields, such as
classification [11], image processing [12], game theory [13] and pest control [14].
To avoid a stalemate when comparing interval data, Bustince et al. [15] introduced the concept
of admissible orders for intervals, that is, total order relations that refine the usual product order
[16], which is a partial order. Since their introduction, several works were developed taking
admissible orders into account, such as [17].
Qiao and Hu [18] and Bedregal et al. [19] defined, independently, the concept of iv-overlap
functions. By extending and generalizing iv-overlap functions, Asmus et al. [11] introduced the
concepts of n-dimensional iv-overlap functions and general iv-overlap functions, both notions
taking into account the usual increasingness with respect to the product order.
Allowing for a broader practical application of (𝑛-dimensional) iv-overlap functions, Asmus et
al. [17] introduced the concept of n-dimensional admissibly ordered iv-overlap functions, which
are n-dimensional iv-overlap functions that are increasing with respect to an admissible order.
They also presented a construction method, which, however, cannot generate n-dimensional
iv-overlap functions that are increasing with respect to the well known lexicographical orders
[15]. Although this is not a serious problem, with the initial motivation to overcome this
drawback, in this present work we combine the recent developed concepts on (𝑛-dimensional,
general) iv-overlap functions and admissible orders to introduce general admissibly ordered
iv-overlap function. The resulting definition proved to be much more flexible and adaptable,
allowing for the development of different construction methods, and even the composition of
functions constructed through those methods.
The paper is organized as follows. Section 2 presents some preliminary concepts. In Section
3, we introduce the concept of general admissibly ordered iv-overlap functions, studying its
representation and some construction methods. Section 4 is the Conclusion.
2. Preliminaries
In this section, we recall some basic concepts used in this work.
An aggregation function [20] is a mapping 𝐴 : [0, 1]𝑛 → [0, 1] that is increasing in each
argument and satisfying: (A1) 𝐴(0, . . . , 0) = 0; (A2) 𝐴(1, . . . , 1) = 1. A function 𝑂𝑛 :
[0, 1]𝑛 → [0, 1] is said to be an n-dimensional
∏︀𝑛overlap function [7] if, for all ⃗𝑥 ∈ [0,
∏︀𝑛1] : (On1)
𝑛
𝑂𝑛 is commutative; (On2) 𝑂𝑛(𝑥 ⃗ ) = 0 ⇔ 𝑖=1 𝑥𝑖 = 0; (On3) 𝑂𝑛(𝑥 ⃗ ) = 1 ⇔ 𝑖=1 𝑥𝑖 = 1;
(On4) 𝑂𝑛 is increasing; (On5) 𝑂𝑛 is continuous. When 𝑂𝑛 is strictly increasing in (0, 1], it is
called a strict n-dimensional overlap function. A 2-dimensional overlap function is just called
overlap function [1, 21].
Definition 1. [8] A function 𝐺𝑂 : [0, 1]𝑛 → [0, 1] is said∏︀𝑛 to be a general overlap function if,
𝑛
∏︀𝑛 all ⃗𝑥 ∈ [0, 1] : (GO1) 𝑂𝑛 is commutative; (GO2) 𝑖=1 𝑥𝑖 = 0 ⇒ 𝐺𝑂(𝑥
for ⃗ ) = 0 (GO3)
𝑖=1 𝑥 𝑖 = 1 ⇒ 𝐺𝑂(𝑥 ⃗ ) = 1; (GO4) 𝐺𝑂 is increasing; (GO5) 𝐺𝑂 is continuous.
Example 1. The following are all examples of general overlap functions, defined for all ⃗𝑥 ∈ [0, 1]𝑛 :
⃗ ) = 𝑛𝑖=1 𝑥𝑖 ;
∏︀
a) The product 𝐺𝑂𝑃 , given by 𝐺𝑂𝑃 (𝑥
⃗ ) = max {( 𝑛𝑖=1 𝑥𝑖 ) − (𝑛 − 1), 0};
∑︀
b) The function 𝐺𝑂𝐿 , given by 𝐺𝑂𝐿 (𝑥
⃗ ) = 𝑛 𝑛𝑖=1 𝑥𝑖 .
√︀∏︀
c) The geometric mean 𝐺𝑂𝐺𝑚 , given by 𝐺𝑂𝐺𝑚 (𝑥
Now, let us denote as 𝐿([0, 1]) the set of all closed subintervals of the unit interval [0, 1].
Denote ⃗𝑥 = (𝑥1 , . . . , 𝑥𝑛 ) ∈ [0, 1]𝑛 and 𝑋
⃗ = (𝑋1 , . . . , 𝑋𝑛 ) ∈ 𝐿([0, 1])𝑛 . Given any 𝑋 =
[𝑥1 , 𝑥2 ] ∈ 𝐿([0, 1]), 𝑋 = 𝑥1 and 𝑋 = 𝑥2 denote, respectively, the left and right projections of
𝑋, and 𝑤(𝑋) = 𝑋 − 𝑋 denotes the width of 𝑋. The interval product order is defined for all
𝑋, 𝑌 ∈ 𝐿([0, 1]) by: 𝑋 ≤𝑃 𝑟 𝑌 ⇔ 𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑌 . We call as ≤𝑃 𝑟 -increasing a function
that is increasing with respect to the product order ≤𝑃 𝑟 .
Given 𝑓, 𝑔 : [0, 1]𝑛 → [0, 1] such that 𝑓 ≤ 𝑔, we define the function 𝑓, ̂︂𝑔 : 𝐿([0, 1])𝑛 →
𝐿([0, 1]) as
𝑓, ⃗ ) = [𝑓 (𝑋1 , . . . , 𝑋𝑛 ), 𝑔(𝑋1 , . . . , 𝑋𝑛 )].
̂︂𝑔(𝑋 (1)
Definition 2. [10] Let 𝐼𝐹 : 𝐿([0, 1])𝑛 → 𝐿([0, 1]) be an ≤𝑃 𝑟 -increasing interval function. 𝐼𝐹 is said to
be representable if there exist increasing functions 𝑓, 𝑔 : [0, 1]𝑛 → [0, 1] such that 𝑓 ≤ 𝑔 and 𝐹 = 𝑓,
̂︂𝑔.
The functions 𝑓 and 𝑔 are the representatives of the interval function 𝐹 . When 𝐹 = 𝑓, ̂︂𝑓 , we denote
simply as 𝑓 .
̂︀
The notion of admissible orders for intervals came from the interest in refining the product order ≤𝑃 𝑟
to a total order.
Definition 3. [15] Let (𝐿([0, 1]), ≤𝐴𝐷 ) be a partially ordered set. The order ≤𝐴𝐷 is an admissible order if:
(i) ≤𝐴𝐷 is a total order on (𝐿([0, 1]), ≤𝐴𝐷 ); (ii) For all 𝑋, 𝑌 ∈ 𝐿([0, 1]), 𝑋 ≤𝐴𝐷 𝑌 whenever 𝑋 ≤𝑃 𝑟 𝑌 .
Example 2. Examples of admissible orders on 𝐿([0, 1]) are the lexicographical orders with respect to the
first and second coordinate, defined, respectively, by:
𝑋 ≤𝐿𝑒𝑥1 𝑌 ⇔ 𝑋 < 𝑌 ∨ (𝑋 = 𝑌 ∧ 𝑋 ≤ 𝑌 ), 𝑋 ≤𝐿𝑒𝑥2 𝑌 ⇔ 𝑋 < 𝑌 ∨ (𝑋 = 𝑌 ∧ 𝑋 ≤ 𝑌 ).
Definition 4. [15] For 𝛼, 𝛽 ∈ [0, 1] such that 𝛼 ̸= 𝛽, the relation ≤𝛼,𝛽 is defined by
𝑋 ≤𝛼,𝛽 𝑌 ⇔ 𝐾𝛼 (𝑋, 𝑋) < 𝐾𝛼 (𝑌 , 𝑌 ) or (𝐾𝛼 (𝑋, 𝑋) = 𝐾𝛼 (𝑌 , 𝑌 ) and 𝐾𝛽 (𝑋, 𝑋) ≤ 𝐾𝛽 (𝑌 , 𝑌 )),
where 𝐾𝛼 , 𝐾𝛽 : [0, 1]2 → [0, 1] are aggregation functions defined, respectively, by
𝐾𝛼 (𝑥, 𝑦) = 𝑥 + 𝛼 · (𝑦 − 𝑥), 𝐾𝛽 (𝑥, 𝑦) = 𝑥 + 𝛽 · (𝑦 − 𝑥). (2)
Then, the relation ≤𝛼,𝛽 is an admissible order.
Remark 1. By varying the values of 𝛼 and 𝛽 one can recover some of the known admissible orders, e.g.,
the lexicographical orders ≤𝐿𝑒𝑥1 and ≤𝐿𝑒𝑥2 can be recovered by ≤0,1 and ≤1,0 , respectively.
For simplicity, we denote 𝐾𝛼 (𝑋, 𝑋) simply as 𝐾𝛼 (𝑋). Also, we denote an iv-function that is
increasing with respect to an admissible order ≤𝐴𝐷 as ≤𝐴𝐷 -increasing. Every ≤𝐴𝐷 -increasing function
is also ≤𝑃 𝑟 -increasing, since every admissible order ≤𝐴𝐷 refines ≤𝑃 𝑟 .
The interval-product is defined, for all 𝑋, 𝑌 ∈ 𝐿([0, 1]), by 𝑋 · 𝑌 = [𝑋 · 𝑌 , 𝑋 · 𝑌 ].
A function 𝐼𝐴 : 𝐿([0, 1])𝑛 → 𝐿([0, 1]) is an iv-aggregation function [22] if: (IA1) 𝐼𝐴 is ≤𝑃 𝑟 -
increasing; (IA2) 𝐼𝐴([0, 0], . . . , [0, 0]) = [0, 0] and 𝐼𝐴([1, 1], . . . , [1, 1]) = [1, 1].
Definition 5. [11] A function 𝐼𝑂𝑛 : 𝐿([0, 1])𝑛 → 𝐿([0, 1]) is an n-dimensional iv-overlap function if, for
⃗ ∈ 𝐿([0, 1])𝑛 : (IOn1) 𝐼𝑂𝑛 is commutative; (IOn2) 𝐼𝑂𝑛(𝑋
all 𝑋 ⃗ ) = [0, 0] ⇔ ∏︀𝑛 𝑋𝑖 = [0, 0]; (IOn3)
𝑖=1
𝐼𝑂𝑛(𝑋 ⃗ ) = [1, 1] ⇔ 𝑛 𝑋𝑖 = [1, 1]; (IOn4) 𝐼𝑂𝑛 is ≤𝑃 𝑟 -increasing; (IOn5) 𝐼𝑂𝑛 is Moore continuous.
∏︀
𝑖=1
For 𝑛 = 2, 𝐼𝑂𝑛 is just called iv-overlap function [18, 19].
Let 𝑂𝑛1 , 𝑂𝑛2 : [0, 1]𝑛 → [0, 1] be 𝑛-dimensional overlap functions such that 𝑂𝑛1 ≤ 𝑂𝑛2 . By [11],
the function 𝐼𝑂𝑛 : 𝐿([0, 1])𝑛 → 𝐿([0, 1]) given, for all 𝑋⃗ ∈ 𝐿([0, 1])𝑛 , by 𝐼𝑂𝑛(𝑋
⃗ ) = 𝑂𝑛ˆ︂ ⃗
1 , 𝑂𝑛2 (𝑋 ),
is a representable 𝑛-dimensional iv-overlap function. As both its representatives are 𝑛-dimensional
overlap functions, it is said to be 𝑜-representable [11].
Definition 6. [11] A function 𝐼𝐺𝑂 : 𝐿([0, 1])𝑛 → 𝐿([0, 1]) is a general iv-overlap function if, for all
⃗ ∈ 𝐿([0, 1])𝑛 : (IGO1) 𝐼𝐺𝑂 is commutative; (IGO2)∏︀𝑛 𝑋𝑖 = [0, 0] ⇒ 𝐼𝐺𝑂(𝑋
𝑋 ⃗ ) = [0, 0]; (IGO3)
∏︀𝑛 𝑖=1
⃗
𝑖=1 𝑋𝑖 = [1, 1] ⇒ 𝐼𝐺𝑂(𝑋 ) = [1, 1]; (IGO4) 𝐼𝐺𝑂 is ≤𝑃 𝑟 -increasing; (IGO5) 𝐼𝐺𝑂 is Moore continuous.
Definition 7. [17] A function 𝐴𝑂𝑛 : 𝐿([0, 1])𝑛 → 𝐿([0, 1]) is an n-dimensional admissibly ordered iv-
overlap function for an admissible order ≤𝐴𝐷 (n-dimensional ≤𝐴𝐷 -overlap function) if it satisfies (IOn1),
(IOn2) and (IOn3) from Def. 5, and (AOn4): 𝐴𝑂𝑛 is ≤𝐴𝐷 -increasing.
Remark 2. Observe that condition (IOn5) was not considered in Def. 7, as the continuity condition of
overlap functions was only a requirement in order for them to be applied in image processing problems,
which was not the case in [17].
The following Theorem presents a construction method for 𝑛-dimensional ≤𝛼,𝛽 -overlap functions:
Theorem 1. [17] Let 𝑂𝑛 be a strict n-dimensional overlap function, 𝛼 ∈ (0, 1) and 𝛽 ∈ [0, 1] such that
⃗ ∈ 𝐿([0, 1])𝑛 , by
𝛼 ̸= 𝛽. Then 𝐴𝑂𝑛𝛼 : 𝐿([0, 1])𝑛 → 𝐿([0, 1]) defined, for all 𝑋
⃗ ) = [𝑂𝑛(𝐾𝛼 (𝑋1 ), . . . , 𝐾𝛼 (𝑋𝑛 )) − 𝛼𝑚, 𝑂𝑛(𝐾𝛼 (𝑋1 ), . . . , 𝐾𝛼 (𝑋𝑛 )) + (1 − 𝛼)𝑚], where
𝐴𝑂𝑛𝛼 (𝑋
𝑚 = min{𝑋1 − 𝑋1 , . . . , 𝑋𝑛 − 𝑋𝑛 , 𝑂𝑛(𝐾𝛼 (𝑋1 ), . . . , 𝐾𝛼 (𝑋𝑛 )), 1 − 𝑂𝑛(𝐾𝛼 (𝑋1 ), . . . , 𝐾𝛼 (𝑋𝑛 ))},
is an 𝑛-dimensional ≤𝛼,𝛽 -overlap function.
Remark 3. Notice that (IOn2) and (IOn3) are both necessary and sufficient conditions. For that reason, the
construction method presented in Theo. 1 must consider 𝛼 ∈ (0, 1) and, consequently, cannot be applied to
obtain neither an 𝑛-dimensional ≤0,1 -overlap function nor an 𝑛-dimensional ≤1,0 -overlap function, that is,
𝑛-dimensional admissibly ordered iv-overlap functions that are increasing with respect to the lexicographical
orders ≤𝐿𝑒𝑥1 and ≤𝐿𝑒𝑥2 , respectively. This drawback is going to be addressed in our developments in
this work. Furthermore, the chosen 𝑛-dimensional overlap function 𝑂𝑛 must be strict, to ensure that the
constructed function is ≤𝛼,𝛽 -increasing.
Let 𝑐 ∈ [0, 1] and 𝛼 ∈ [0, 1]. Denote by 𝑑𝛼 (𝑐) the maximal possible width of an interval 𝑍 ∈ 𝐿([0, 1])
𝑤(𝑋)
such that 𝐾𝛼 (𝑍) = 𝑐. For any 𝑋 ∈ 𝐿([0, 1]), define 𝜆𝛼 (𝑋) = 𝑑𝛼 (𝐾 𝛼 (𝑋))
. In [23], it was shown that
{︂ }︂
𝐾𝛼 (𝑋) 1 − 𝐾𝛼 (𝑋)
𝑑𝛼 (𝐾𝛼 (𝑋)) = min , ,
𝛼 1−𝛼
where it is set that 0𝑟 = 1, for all 𝑟 ∈ [0, 1].
Theorem 2. [23] Let 𝛼, 𝛽 ∈ [0, 1], such that, 𝛼 ̸= 𝛽. Let 𝐴1 , 𝐴2 : [0, 1]𝑛 → [0, 1] be two aggregation
functions where 𝐴1 is strictly increasing. Then 𝐼𝐹 𝛼 : 𝐿([0, 1])𝑛 → 𝐿([0, 1]) defined by:
{︂
𝛼
𝐼𝐹𝐴1,𝐴2 ⃗ ) = 𝑅, where, 𝐾𝛼 (𝑅) = 𝐴1 (𝐾𝛼 (𝑋1 ), . . . , 𝐾𝛼 (𝑋𝑛 )),
(𝑋
𝜆𝛼 (𝑅) = 𝐴2 (𝜆𝛼 (𝑋1 ), . . . , 𝜆𝛼 (𝑋𝑛 )),
⃗ ∈ 𝐿([0, 1])𝑛 , is an ≤𝛼,𝛽 -increasing iv-aggregation function.
for all 𝑋
Corollary 1. Let 𝛼, 𝛽 ∈ [0, 1], 𝛼 ̸= 𝛽. Let 𝑂𝑛 : [0, 1]𝑛 → [0, 1] be a strict 𝑛-dimensional overlap function
and 𝐴 : [0, 1]𝑛 → [0, 1] be an aggregation function. Then 𝐼𝐹𝑂,𝐴
𝛼
: 𝐿([0, 1])𝑛 → 𝐿([0, 1]) defined by:
{︂
𝛼 ⃗ 𝐾𝛼 (𝑅) = 𝑂𝑛(𝐾𝛼 (𝑋1 ), . . . , 𝐾𝛼 (𝑋𝑛 )),
𝐼𝐹𝑂𝑛,𝐴 (𝑋 ) = 𝑅, where,
𝜆𝛼 (𝑅) = 𝐴(𝜆𝛼 (𝑋1 ), . . . , 𝜆𝛼 (𝑋𝑛 )),
⃗ ∈ 𝐿([0, 1])𝑛 , is an ≤𝛼,𝛽 -increasing iv-aggregation function.
for all 𝑋
3. General admissibly ordered iv-overlap functions
By combining the concepts of general iv-overlap functions and 𝑛-dimensional admissibly ordered iv-
overlap functions, we introduce the following definition:
Definition 8. A function 𝐴𝐺𝑂 : 𝐿([0, 1])𝑛 → 𝐿([0, 1]) is a general admissibly ordered iv- overlap
function for an admissible order ≤𝐴𝐷 (general ≤𝐴𝐷 -overlap function) if it satisfies the conditions (IGO1),
(IGO2) and (IGO3) of Def. 1, and (AGO4): 𝐴𝐺𝑂 is ≤𝐴𝐷 -increasing.
The following result is immediate:
Proposition 1. If 𝐴𝑂𝑛 : 𝐿([0, 1])𝑛 → 𝐿([0, 1]) is an 𝑛-dimensional ≤𝐴𝐷 -overlap function, then it is
also a general ≤𝐴𝐷 -overlap function, but the converse may not hold.
Here we present some results regarding representable general iv-overlap functions and their increas-
ingness with respect to a particular admissible order. In the following result, consider that a strict general
overlap function is a general overlap function that is strictly increasing in (0, 1].
Lemma 1. Let 𝐺𝑂 : [0, 1]𝑛 → [0, 1] be a strict general overlap function. Then, 𝐺𝑂 is an 𝑛-dimensional
overlap function.
Proof. It is immediate that 𝐺𝑂 satisfies (On1), (On4) and (On5) and, by (GO2) and (GO3), it respects
the necessary conditions (⇐) of (On2) and (On3). Then, we prove the sufficient conditions.
(On2) (⇒) Suppose that 𝐺𝑂 is strict and does not respect (On2) (⇒). Take ⃗𝑦 ∈ (0, 1]𝑛 such that
𝐺𝑂(𝑦 ⃗ ) = 0. Then, there exist ⃗𝑥 ∈ (0, 1]𝑛 such that ⃗𝑥 < ⃗𝑦 and, by (GO4), 𝐺𝑂(𝑥 ⃗ ) = 𝐺𝑂(𝑦 ⃗ ) = 0, which is
a contradiction since 𝐺𝑂 is strict. Thus, 𝐺𝑂 respects (On2).
(On3) (⇒) Suppose that 𝐺𝑂 is strict and does not respect (On3) (⇒). By (GO2), one has that ⃗𝑥 =
(1, . . . , 1) ⇒ 𝐺𝑂(𝑥⃗ ) = 1. Now, take ⃗𝑦 ∈ [0, 1]𝑛 such that 𝑦𝑖 ̸= 1 for some 𝑖 ∈ {1, . . . , 𝑛} and 𝐺𝑂(𝑦 ⃗ ) = 1.
Then, one has that ⃗𝑦 < ⃗𝑥 and 𝐺𝑂(𝑦 ⃗ ) = 𝐺𝑂(𝑥 ⃗ ) = 1, which is a contradiction since 𝐺𝑂 is strict. Thus, 𝐺𝑂
respects (On3).
Theorem 3. Let 𝐼𝐺𝑂 : 𝐿([0, 1])𝑛 → 𝐿([0, 1]) be a representable general iv-overlap function such that
𝐼𝐺𝑂 = 𝑓, ̂︂𝑔, with 𝑓, 𝑔 : [0, 1]𝑛 → [0, 1] and 𝛼, 𝛽 ∈ [0, 1], 𝛼 ̸= 𝛽. Then, 𝐼𝐺𝑂 is ≤𝛼,𝛽 -increasing if and
only if 𝛼 = 1 and 𝑔 is a strict n-dimensional overlap function.
Proof. Analogous to the proof of Theo. 3 in [17], taking into account Lem. 1.
Then, the following result is immediate:
Corollary 2. Let 𝑂𝑛 : [0, 1]𝑛 → [0, 1] be an 𝑛-dimensional overlap function and 𝐼𝐺𝑂 : 𝐿([0, 1])𝑛 →
̂︁ and 𝛼, 𝛽 ∈ [0, 1], 𝛼 ̸= 𝛽. Then, 𝐼𝐺𝑂 is a
𝐿([0, 1]) be a general iv-overlap function such that 𝐼𝐺𝑂 = 𝑂𝑛,
general ≤𝛼,𝛽 -overlap if and only if 𝛼 = 1 and 𝑂𝑛 is a strict n-dimensional overlap function.
Example 3. Consider the general overlap function 𝐺𝑂𝑃 as defined in Ex. 1 for 𝑛 = 2. As it is a strict
general overlap function, then, by Lem. 1, it is also a strict overlap function. Then, the iv-function 𝐴𝐺𝑂𝑃 :
𝐿([0, 1])2 → 𝐿([0, 1]) defined, for all 𝑋⃗ ∈ 𝐿([0, 1])2 , by 𝐴𝐺𝑂𝑃 (𝑋 ⃗ ) = 𝐺𝑂ˆ︁ ⃗
𝑃 (𝑋 ) is a general ≤1,0 -
overlap function, and also an 2-dimensional ≤1,0 -overlap function.
The first construction method for general ≤𝐴𝐷 -overlap functions is an adaptation of Theo. 1, by taking
𝛼 ∈ [0, 1], obtaining a general ≤𝛼,𝛽 -overlap function.
Theorem 4. Let 𝑂𝑛 be a strict n-dimensional overlap function, 𝛼, 𝛽 ∈ [0, 1] such that 𝛼 ̸= 𝛽. Then
⃗ ∈ 𝐿([0, 1])𝑛 , by
𝐴𝐺𝑂𝛼 : 𝐿([0, 1])𝑛 → 𝐿([0, 1]) defined, for all 𝑋
⃗ ) = [𝑂𝑛(𝐾𝛼 (𝑋1 ), . . . , 𝐾𝛼 (𝑋𝑛 )) − 𝛼𝑚, 𝑂𝑛(𝐾𝛼 (𝑋1 ), . . . , 𝐾𝛼 (𝑋𝑛 )) + (1 − 𝛼)𝑚], where
𝐴𝐺𝑂𝛼 (𝑋
𝑚 = min{𝑋1 − 𝑋1 , . . . , 𝑋𝑛 − 𝑋𝑛 , 𝑂𝑛(𝐾𝛼 (𝑋1 ), . . . , 𝐾𝛼 (𝑋𝑛 )), 1 − 𝑂𝑛(𝐾𝛼 (𝑋1 ), . . . , 𝐾𝛼 (𝑋𝑛 ))},
is a general ≤𝛼,𝛽 -overlap function.
Proof. Analogous to the proof of Theo. 4 in [17].
Remark 4. Observe that (IGO2) and (IGO3) are only sufficient conditions, allowing for 𝛼 ∈ [0, 1] in the
construction method presented in Theo. 4, differently than in Theo. 1, in which 𝛼 ∈ (0, 1). This means that,
through Theo. 4, one can obtain general ≤𝐴𝐷 -overlap functions that are increasing with respect to either
one of the lexicographical orders.
Remark 5. Regarding Theo. 4, one could think that it could be based on a general overlap function 𝐺𝑂
instead of a 𝑛-dimensional overlap function 𝑂𝑛, for it to be even more broad of a method. However, as
the base function needs to be strictly increasing in order to the constructed iv-function 𝐴𝐺𝑂𝛼 to be ≤𝛼,𝛽 -
increasing, by Lem. 1, one has that every strict general overlap function is also an 𝑛-dimensional overlap
function, and that is why we chose to maintain 𝑂𝑛 in Theo. 4 to reinforce this fact.
Example 4. Consider the general overlap function 𝐺𝑂𝑃 as defined in Ex. 1. Then, for 𝛼 = 1 and 𝛽 = 0,
⃗ ∈ 𝐿([0, 1])𝑛 , by
the iv-function 𝐴𝐺𝑂𝑃1 : 𝐿([0, 1])𝑛 → 𝐿([0, 1]) defined for all 𝑋
⃗ ) = [𝐺𝑂𝑃 (𝑋1 , . . . , 𝑋𝑛 ) − 𝑚, 𝐺𝑂𝑃 (𝑋1 , . . . , 𝑋𝑛 )], where
𝐴𝐺𝑂𝑃1 (𝑋
𝑚 = min{𝑋1 − 𝑋1 , . . . , 𝑋𝑛 − 𝑋𝑛 , 𝐺𝑂𝑃 (𝑋1 , . . . , 𝑋𝑛 ), 1 − 𝐺𝑂𝑃 (𝑋1 , . . . , 𝑋𝑛 )},
is a general ≤1,0 -overlap function, or in other words, a general ≤𝐿𝑒𝑥2 -overlap function. It is noteworthy
that 𝐴𝐺𝑂𝑃1 is not an 𝑛-dimensional ≤1,0 -overlap function.
The next construction methods are inspired on Theo. 2. First, we will present a more restrictive construction
method for 𝑛-dimensional ≤𝛼,𝛽 -overlap functions:
Theorem 5. Let 𝛼, 𝛽 ∈ (0, 1), 𝛼 ̸= 𝛽. Let 𝑂𝑛 : [0, 1]𝑛 → [0, 1] be a strict 𝑛-dimensional overlap function
and 𝐴 : [0, 1]𝑛 → [0, 1] a commutative aggregation function. 𝐴𝑂𝑛𝛼 𝑛
𝐴 : 𝐿([0, 1]) → 𝐿([0, 1]) defined by
{︂
⃗ 𝐾𝛼 (𝑅) = 𝑂𝑛(𝐾𝛼 (𝑋1 ), . . . , 𝐾𝛼 (𝑋𝑛 )),
𝐴𝑂𝑛𝛼 𝐴 (𝑋 ) = 𝑅, where,
𝜆𝛼 (𝑅) = 𝐴(𝜆𝛼 (𝑋1 ), . . . , 𝜆𝛼 (𝑋𝑛 )),
⃗ ∈ 𝐿([0, 1])𝑛 , is an 𝑛-dimensional ≤𝛼,𝛽 -overlap function.
for all 𝑋
Proof. From Theo. 2, it is immediate that 𝐴𝑂𝑛𝛼 𝐴 is well defined and ≤𝛼,𝛽 -increasing, thus, respecting
condition (AOn4). Now, let us verify if 𝐴𝑂𝑛𝛼 𝐴 respects the remainder conditions from Def. 7:
(IOn1) Immediate, since 𝑂𝑛 and 𝐴 are commutative.
(IOn2) (⇒) Take 𝑋 ⃗ ∈ 𝐿([0, 1])𝑛 and suppose that 𝐴𝑂𝑛𝛼 (𝑋 ⃗ ) = 𝑅 = [0, 0]. Then, we have that
𝐴
𝐾𝛼 (𝑅) = 𝐾𝛼 ([0, 0]) = 0 = 𝑂𝑛(𝐾𝛼 (𝑋1 ), . . . , 𝐾𝛼 (𝑋𝑛 )), for all 𝛼 ∈ (0, ∏︀𝑛1). Thus, by condition (On2),
𝐾𝛼 (𝑋𝑖 ) = 0 for some 𝑖 ∈ {1, . . . , 𝑛}, for all 𝛼 ∈ (0, 1), and, therefore, 𝑖=1 𝑋𝑖 = [0, 0]. (⇐) Consider
𝑋⃗ ∈ 𝐿([0, 1])𝑛 such that ∏︀𝑛 = [0, 0]. So, 𝐾𝛼 (𝑋1 ) · . . . · 𝐾𝛼 (𝑋𝑛 ) = 0, for all 𝛼 ∈ (0, 1). Then,
𝑖=1
by (On2), one has that 𝐾𝛼 (𝑅) = 𝑂𝑛(𝐾𝛼 (𝑋1 ), . . . , 𝐾𝛼 (𝑋𝑛 )) = 0, for all 𝛼 ∈ (0, 1), meaning that
𝐴𝑂𝑛𝛼 ⃗
𝐴 (𝑋 ) = 𝑅 = [0, 0];
(IOn3) (⇒) Take 𝑋 ⃗ ∈ 𝐿([0, 1])𝑛 such that 𝐴𝑂𝑛𝛼 (𝑋 ⃗ ) = 𝑅 = [1, 1]. Then, one has that 𝐾𝛼 (𝑅) =
𝐴
𝐾𝛼 ([1, 1]) = 1 = 𝑂𝑛(𝐾𝛼 (𝑋1 ), . . . , 𝐾𝛼 (𝑋𝑛 )). By (On3), 𝐾𝛼 (𝑋1 ) · . . . · 𝐾𝛼 (𝑋𝑛 ) = 1, for all 𝛼 ∈
∏︀𝑛 ⃗ ∈ 𝐿([0, 1])𝑛 such that ∏︀𝑛 𝑋𝑖 = [1, 1].
(0, 1), meaning that 𝑖=1 𝑋𝑖 = [1, 1]. (⇐) Consider 𝑋 𝑖=1
So, 𝐾𝛼 (𝑋1 ) · . . . · 𝐾𝛼 (𝑋𝑛 ) = 1, for all 𝛼 ∈ (0, 1). Then, by (i) and (O3), one has that 𝐾𝛼 (𝑅) =
𝑂𝑛(𝐾𝛼 (𝑋1 ), . . . , 𝐾𝛼 (𝑋𝑛 )) = 1, for all 𝛼 ∈ (0, 1), meaning that 𝐴𝑂𝑛𝛼 ⃗
𝐴 (𝑋 ) = 𝑅 = [1, 1].
The following result is immediate, as it derives from a similar situation as discussed in Remarks 4 and 5.
Theorem 6. Let 𝛼, 𝛽 ∈ [0, 1], 𝛼 ̸= 𝛽. Let 𝑂𝑛 : [0, 1]𝑛 → [0, 1] be a strict 𝑛-dimensional overlap function
and 𝐴 : [0, 1]𝑛 → [0, 1] a commutative aggregation function. 𝐴𝐺𝑂𝐴 𝛼
: 𝐿([0, 1])𝑛 → 𝐿([0, 1]) defined by
{︂
𝛼 ⃗ 𝐾𝛼 (𝑅) = 𝑂𝑛(𝐾𝛼 (𝑋1 ), . . . , 𝐾𝛼 (𝑋𝑛 )),
𝐴𝐺𝑂𝐴 (𝑋 ) = 𝑅, where,
𝜆𝛼 (𝑅) = 𝐴(𝜆𝛼 (𝑋1 ), . . . , 𝜆𝛼 (𝑋𝑛 )),
⃗ ∈ 𝐿([0, 1])𝑛 , is an general ≤𝛼,𝛽 -overlap function.
for all 𝑋
Example 5. Consider the general overlap functions 𝐺𝑂𝐿 and 𝐺𝑂𝐺𝑚 as defined in Ex. 1. For 𝛼 = 1 and
⃗ ∈ 𝐿([0, 1])𝑛 , by
𝛽 = 0, the iv- function 𝐴𝐺𝑚1𝐺𝑂𝐿 : 𝐿([0, 1])𝑛 → 𝐿([0, 1]) defined for all 𝑋
{︂
⃗ )=𝑅, 𝐾1 (𝑅)=𝐺𝑂𝐺𝑚 (𝑋1 , . . ., 𝑋𝑛 ),
𝐴𝐺𝑚1𝐺𝑂𝐿 (𝑋 where,
𝜆1 (𝑅)=𝐺𝑂𝐿 (𝜆1 (𝑋1 ), . . ., 𝜆1 (𝑋𝑛 )),
is a general ≤1,0 -overlap function, but not an 𝑛-dimensional ≤1,0 -overlap function.
The following method allow the construction of general ≤𝐴𝐷 -overlap functions by the generalized
composition of general ≤𝐴𝐷 -overlap functions by an ≤𝐴𝐷 -increasing iv-aggregation function.
−−−→
Theorem 7. Consider 𝐼𝑀 : 𝐿([0, 1])𝑚 → 𝐿([0, 1]). For a tuple 𝐴𝐺𝑂 = (𝐴𝐺𝑂1 , . . . , 𝐴𝐺𝑂𝑚 ) of general
≤𝐴𝐷 -overlap functions, define the mapping 𝐼𝑀− ⃗ ∈ 𝐿([0, 1])𝑛 , by:
−−→ : 𝐿([0, 1])𝑛 → 𝐿([0, 1]), for all 𝑋
𝐴𝐺𝑂
𝐼𝑀− −−→ (𝑋⃗ ) = 𝐼𝑀 (𝐴𝐺𝑂1 (𝑋 ⃗ ), . . . , 𝐴𝐺𝑂𝑚 (𝑋
⃗ )). Then, 𝐼𝑀−−−→ is a general ≤𝐴𝐷 -overlap function if
𝐴𝐺𝑂 𝐴𝐺𝑂
and only if 𝐼𝑀 is an ≤𝐴𝐷 -increasing iv-aggregation function.
Proof. (⇒) Suppose that 𝐼𝑀− −−→ is a general ≤𝐴𝐷 -overlap function. Then it is immediate that 𝐼𝑀
𝐴𝐺𝑂
≤𝐴𝐷 -increasing, and, also, ≤𝑃 𝑟 -increasing (IA2). Now consider 𝑋 ⃗ ∈ 𝐿([0, 1])𝑛 such that ∏︀𝑛 𝑋𝑖 =
𝑖=1
[0, 0]. Then, by (IGO2), one has that: 𝐼𝑀− −−→ (𝑋⃗ ) = 𝐼𝑀 (𝐴𝐺𝑂1 (𝑋 ⃗ ), . . . , 𝐴𝐺𝑂𝑚 (𝑋 ⃗ )) = [0, 0] and
𝐴𝐺𝑂
𝐴𝐺𝑂1 (𝑋 ⃗ ) = . . . = 𝐴𝐺𝑂𝑚 (𝑋 ⃗ ) = [0, 0]. Thus, it holds that 𝐼𝑀 ([0, 0], . . . , [0, 0]) = [0, 0]. Now,
consider 𝑋 ⃗ ∈ 𝐿([0, 1]) , such that 𝑋𝑖 = [1, 1] for all 𝑖 ∈ {1, . . . , 𝑛}. Then, by (IGO3), one has that:
𝑛
𝐼𝑀𝐴𝐺𝑂 (𝑋
−−−→ ⃗ ) = 𝐼𝑀 (𝐴𝐺𝑂1 (𝑋 ⃗ ), . . . , 𝐴𝐺𝑂𝑚 (𝑋⃗ )) = [1, 1] and 𝐴𝐺𝑂1 (𝑋 ⃗ ) = . . . = 𝐴𝐺𝑂𝑚 (𝑋 ⃗ ) = [1, 1].
Therefore, it holds that 𝐼𝑀 ([1, 1], . . . , [1, 1]) = [1, 1]. This proves that 𝐼𝑀 also satisfies condition (IA1),
and, thus, an ≤𝐴𝐷 -increasing iv-aggregation function. (⇐) Suppose that 𝐼𝑀 is an ≤𝐴𝐷 -increasing iv-
aggregation function. Then it is immediate that 𝐼𝑀− −−→ is commutative (by (IGO1)), and respects (AGO4).
𝐴𝐺𝑂
⃗ ∈ 𝐿([0, 1]) such that
𝑛
∏︀ 𝑛 ⃗
(IGO2) Consider 𝑋 𝑖=1 𝑋𝑖 = [0, 0]. Then, by (IGO2), one has that 𝐴𝐺𝑂1 (𝑋 ) =
⃗ ⃗
. . . = 𝐴𝐺𝑂𝑚 (𝑋 ) = [0, 0]. It follows that: 𝐼𝑀𝐴𝐺𝑂 (𝑋 ) = 𝐼𝑀 (𝐴𝐺𝑂1 (𝑋 ), . . . , 𝐴𝐺𝑂𝑚 (𝑋 )) =
−
− −→ ⃗ ⃗
𝐼𝑀 ([0, 0], . . . , [0, 0]) = [0, 0], by condition (IA1), since 𝐼𝑀 is an iv-aggregation function.
(IGO3) Take 𝑋 ⃗ ∈ 𝐿([0, 1])𝑛 such that 𝑋𝑖 = [1, 1] for all 𝑖 ∈ {1, . . . , 𝑛}. Then, by (IGO3), it holds that
𝐴𝐺𝑂1 (𝑋 ⃗ ) = . . . = 𝐴𝐺𝑂𝑚 (𝑋 ⃗ ) = [1, 1]. Thus, 𝐼𝑀−−−→ (𝑋 ⃗ ) = 𝐼𝑀 (𝐴𝐺𝑂1 (𝑋 ⃗ ), . . . , 𝐴𝐺𝑂𝑚 (𝑋 ⃗ )) =
𝐴𝐺𝑂
𝐼𝑀 ([1, 1], . . . , [1, 1]) = [1, 1], by (IA1). Then, 𝐼𝑀− ⃗
−−→ (𝑋 ) is a general ≤𝐴𝐷 -overlap function.
𝐴𝐺𝑂
Example 6. Consider the general ≤1,0 -overlap functions 𝐴𝐺𝑂𝑃 , 𝐴𝐺𝑂𝑃1 and 𝐴𝐺𝑚1𝐺𝑂𝐿 , from Ex.s 3, 4
⃗ ∈ 𝐿([0, 1])𝑛 , by 𝐴𝐺𝑂(𝑋
and 5. Then, the iv-function 𝐴𝐺𝑂 : 𝐿([0, 1])𝑛 → 𝐿([0, 1]) defined, for all 𝑋 ⃗)=
1 ⃗ 1 ⃗
𝐴𝐺𝑂𝑃 (𝐴𝐺𝑂𝑃 (𝑋 ), 𝐴𝐺𝑚𝐺𝑂𝐿 (𝑋 )), is a general ≤1,0 -overlap function.
4. Conclusion
In this paper we presented the concept of general admissibly ordered iv-overlap functions, a more flexible
definition of 𝑛-dimensional iv-overlap functions that are increasing with respect to an admissible order. This
new definition allowed us to construct several iv-overlap operations taking into account different admissible
orders, in particular, ≤𝛼,𝛽 orders with any 𝛼, 𝛽 ∈ [0, 1] such that 𝛼 ̸= 𝛽, which includes the lexicographical
orders. Finally, those constructed functions can be combined by generalized composition to obtain new
general admissibly ordered iv-overlap functions, showcasing their adaptability.
Most construction methods for ≤𝛼,𝛽 -increasing functions are based on the aggregation of the 𝐾𝛼 values
of the inputs by strictly increasing aggregation functions, which is a restriction that could be interesting to
overcome in our future work. We also intend to apply the developed functions (with different combination of
construction methods) in classification problems with interval-valued data.
Acknowledgments
Supported by the Spanish Ministries of Science and Technology (PC093-094 TFIPDL, TIN2016-81731-REDT,
TIN2016-77356-P (AEI/FEDER, UE)), and of Economy and Competitiveness (PID2019-108392GB-I00/AEI/
10.13039/501100011033), UPNA (PJUPNA1926), CNPq (301618/2019-4) and FAPERGS (19/2551-0001660).
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