On the Composition of Overlap and Grouping Functions

Abstract

Obtaining overlap/grouping functions from a given pair of overlap/grouping functions is an important method of generating overlap/grouping functions, which can be viewed as a binary operation on the set of overlap/grouping functions. In this paper, firstly, we studied closures of overlap/grouping functions w.r.t. ⊛-composition. In addition, then, we show that these compositions are order preserving. Finally, we investigate the preservation of properties like idempotency, migrativity, homogeneity, k-Lipschitz, and power stable.

1. Introduction

Overlap function [1] is a special case of aggregation functions [2]. Grouping function [3] is the dual concept of overlap function. In recent years, overlap and grouping functions have attracted wide interest. In the field of application, they are used in image processing [1,4], classification [5,6], and decision-making [7,8]. In the field of theoretical research, the concepts of general, Archimedean, n-dimensional, interval-valued, and complex-valued overlap/grouping functions have been introduced [9,10,11,12,13,14,15,16,17]. In the literature about overlap/grouping functions, much attention have been recently paid to their properties, this study has enriched overlap/grouping functions. Bedregal [9] studied some properties such as migrativity, idempotency, and homogeneity of overlap/overlap functions. Gomez et al. [12] also considered these properties of N-dimensional overlap functions. Costa and Bedregal [18] introduced quasi-homogeneous overlap functions. Qian and Hu [19] studied the migrativity of uninorms and nullnorms over overlap/grouping functions. They [13,20,21] also studied multiplicative generators and additive generators of overlap/grouping functions and the distributive laws of fuzzy implication functions over overlap functions [9,12,13,18,19,20,21]. Moreover, overlap/grouping functions also can be viewed as binary connectives on [0, 1], then they can be used to construct other fuzzy connectives. Residual implication, (G, N)-implications, QL-implications, (IO, O)-fuzzy rough sets, and binary relations induced from overlap/grouping functions have been studied [22,23,24,25,26,27].

The construction of the following overlap/grouping functions was developed in many literature works [1,4,13,15,16,21,27,28]. Obtaining overlap/grouping functions from given overlap/grouping functions is one of the methods to generate overlap/grouping functions. We consider this work as a composition of two or more overlap/grouping functions. As mentioned above, some properties are important for overlap/grouping functions. Thus, it raises the question of whether the new generated overlap/grouping function still satisfies the properties of overlap/grouping functions. In this paper, we consider properties preservation of four compositions such as meet operation, join operation, convex combination, and ⊛-composition of overlap/grouping functions. These results might serve as a certain criteria for choices of generation methods of overlap/grouping functions from given overlap/grouping functions.

The paper is organized as follows: In Section 2, we recall the concepts of overlap/grouping functions and their properties. In Section 3, we studied the closures of overlap/grouping functions w.r.t. ⊛-composition. In Section 4, we study the order preservation of compositions. In Section 5, we study properties’ preservation of compositions. In Section 6, conclusions are briefly summed up.

2. Preliminaries

2.1. Overlap and Grouping Functions

First, we recall the concepts of overlap/grouping functions and their properties; for details, see [1,9,12,13].

Definition 1

([1]). A bivariate function $O:{[0,1]}^2\to [0,1]$ is an overlap function if it has the following properties:

(O1) 

It is commutative;

(O2) 

$O(\eta ,\xi )=0$ if and only if $\eta \xi =0$;

(O3) 

$O(\eta ,\xi )=1$ if and only if $\eta \xi =1$;

(O4) 

It is non-decreasing;

(O5) 

It is continuous.

Definition 2

([1]). A bivariate function $G:{[0,1]}^2\to [0,1]$ is a grouping function if it has the following properties:

(G1) 

It is commutative;

(G2) 

$G(\eta ,\xi )=0$ if and only if $\eta =\xi =0$;

(G3) 

$G(\eta ,\xi )=1$ if and only if $\eta =1$ or $\xi =1$.

(G4) 

It is non-decreasing;

(G5) 

It is continuous.

If O is an overlap function, then the function $G(\eta ,\xi )=1-O(1-\eta ,1-\xi )$ is the dual grouping function of G.

2.2. Properties of Overlap and Grouping Functions

For any two overlap (or grouping) functions O and $O^{\prime }$, if $O(\eta ,\xi )\le O^{\prime }(\eta ,\xi )$ holds for all $(\eta ,\xi )\in {[0,1]}^2$, then we say that O is weaker than $O^{\prime }$, denoted $O⪯O^{\prime }$. For example, consider the following three overlap functions $O_M(\eta ,\xi )=min(\eta ,\xi )$, $O_P(\eta ,\xi )=\eta \xi$ and $O_{Mid}(\eta ,\xi )=\eta \xi \frac{\eta +\xi }{2}$, we get this ordering for these overlap functions:

$$O_{Mid}⪯O_P⪯O_M.$$

Some interesting properties for overlap (or grouping) functions are:

(ID) 

Idempotency:

$$O(\eta ,\eta )=\eta$$

for all $\eta \in [0,1]$;

(MI) 

Migrativity:

$$O(\alpha \eta ,\xi )=O(\eta ,\alpha \xi )$$

for all $\alpha ,\eta ,\xi \in [0,1]$;

(HO-k) 

Homogeneous of order $k\in ]0,\infty [$:

$$O(\alpha \eta ,\alpha \xi )={\alpha }^{k}O(\eta ,\xi )$$

for all $\alpha \in [0,\infty [$ and $\eta ,\xi \in [0,1]$ such that $\alpha \eta ,\alpha \xi \in [0,1]$;

(k-LI) 

k-Lipschitz:

$$|O({\eta }_1,{\xi }_1)-O({\eta }_2,{\xi }_2)| \le k(|{\eta }_1-{\eta }_2|+|{\xi }_1-{\xi }_2\left|\right)$$

for all ${\eta }_1,{\eta }_2,{\xi }_1,{\xi }_2\in [0,1]$.

(PS) 

Power stable [29]:

$$O({\eta }^r,{\xi }^r)=O{(\eta ,\xi )}^r$$

for all $r\in ]0,\infty [$ and $\eta ,\xi \in [0,1]$.

3. Compositions of Overlap and Grouping Functions and Their Closures

In the following, we list four compositions of overlap/grouping functions including meet, join, convex combination, and ⊛-composition. In addition, we then studied their closures.

3.1. Compositions of Overlap and Grouping Functions

For any two overlap (or grouping) functions $O_1$ and $O_2$, meet and join operations of $O_1$ and $O_2$ are defined by

$$(O_1\vee O_2)(\eta ,\xi )=max\left(O_1(\eta ,\xi ),O_2(\eta ,\xi )\right),$$

(1)

$$(O_1\wedge O_2)(\eta ,\xi )=min\left(O_1(\eta ,\xi ),O_2(\eta ,\xi )\right)$$

(2)

for all $(\eta ,\xi )\in {[0,1]}^2$.

For any two overlap (or grouping) functions $O_1$ and $O_2$, a convex combination of $O_1$ and $O_2$ is defined as

$$\begin{array}{c}\hfill O_{\lambda }=\lambda O_1(\eta ,\xi )+(1-\lambda )O_2(\eta ,\xi )\end{array}$$

(3)

for all $(\eta ,\xi )\in {[0,1]}^2$ and $\lambda \in [0,1]$.

For any two overlap (or grouping) functions $O_1$ and $O_2$, the ⊛-composition of $O_1$ and $O_2$ is defined as

$$(O_1⊛O_2)(\eta ,\xi )=O_1\left(\eta ,O_2(\eta ,\xi )\right)$$

(4)

for all $(\eta ,\xi )\in {[0,1]}^2$.

3.2. Closures of the Compositions

Closures of the meet operation, join operation, and convex combination have been obtained in [1,3,9]. The ⊛-composition of two overlap functions is closed means ⊛-composition of two bivariate functions on [0, 1] preserves (O1), (O2), (O3), (O4) and (O5). Similarly, the ⊛-composition of two grouping functions is closed means ⊛-composition of two bivariate functions on [0, 1] preserves (G1), (G2), (G3), (G4) and (G5).

Theorem 1.

If two bivariate functions $O_1,O_2:{[0,1]}^2\to [0,1]$ satisfy (O2)((O3), (G2), (G3), (O4), (O5)), then $(O_1⊛O_2)$ also satisfies (O2)((O3), (G2), (G3), (O4), (O5)).

Proof. 

First, we show that ⊛-composition preserves (O2). If

$$(O_1⊛O_2)(\eta ,\xi )=O_1\left(\eta ,O_2(\eta ,\xi )\right)=0,$$

then, since $O_1$ satisfies (O2), we have $\eta O_2(\eta ,\xi )$=0. Case I, if $\eta =0$ and $O_2(\eta ,\xi )\ne 0$, then $\eta \xi =0\xi =0$; Case II, if $\eta =0$ and $O_2(\eta ,\xi )=0$, then $\eta \xi =0\xi =0$; Case III, if $\eta \ne 0$ and $O_2(\eta ,\xi )=0$, since $O_2$ satisfies (O2), then $\eta \xi =0$.

Next, we show that ⊛-composition preserves (O3). If

$$(O_1⊛O_2)(\eta ,\xi )=O_1\left(\eta ,O_2(\eta ,\xi )\right)=1,$$

then, since $O_1$ satisfies (O3), we have $\eta O_2(\eta ,\xi )$=1. Then, $\eta =1$ and $O_2(\eta ,\xi )=1$, since $O_2$ satisfies (O3), then $\eta \xi =1$.

Then, we show that ⊛-composition preserves (G2). If

$$(O_1⊛O_2)(\eta ,\xi )=O_1\left(\eta ,O_2(\eta ,\xi )\right)=0,$$

then, since $O_1$ satisfies (G2), we have $\eta =O_2(\eta ,\xi )$=0. Since $O_2$ satisfies (G2), then $\eta =\xi =0$.

Afterwards, we show that ⊛-composition preserves (G3). If

$$(O_1⊛O_2)(\eta ,\xi )=O_1\left(\eta ,O_2(\eta ,\xi )\right)=1,$$

then, since $O_1$ satisfies (G3), we have $\eta =1$ or $O_2(\eta ,\xi )$=1. Since $O_2$ satisfies (G3), $O_2(\eta ,\xi )$=1 means $\eta =1$ or $\xi =1$.

The case for (O4) and (O5) are straightforward. □

Unfortunately, ⊛-composition of two bivariate functions does not preserve (O1). For example, let $O_1(\eta ,\xi )=O_2(\eta ,\xi )=\eta \xi$; then, $(O_1⊛O_2)(\eta ,\xi )={\eta }^2\xi$ is not commutative. This means ⊛-composition of two overlap/grouping functions is not closed.

However, it is possible to find an example that ⊛-composition of two overlap/grouping functions is also an overlap/grouping function. For example, for two given overlap functions $O_1(\eta ,\xi )=O_2(\eta ,\xi )=min(\eta ,\xi )$, their ⊛-composition $(O_1⊛O_2)(\eta ,\xi )=min(\eta ,\xi )$ is an overlap function.

The summary of the closures of two bivariate functions w.r.t. these compositions is shown in

Table 1. Closures of the compositions.

Property	${\mathit{O}}_1$	${\mathit{O}}_2$	${\mathit{O}}_1\vee {\mathit{O}}_2$	${\mathit{O}}_1\wedge {\mathit{O}}_2$	${\mathit{O}}_{\mathit{\lambda }}$	${\mathit{O}}_1⊛{\mathit{O}}_2$
${\mathit{O}}_{\mathbf{1}}$	√	√	√	√	√	×
${\mathit{O}}_{\mathbf{2}}$	√	√	√	√	√	√
${\mathit{O}}_{\mathbf{3}}$	√	√	√	√	√	√
${\mathit{G}}_{\mathbf{2}}$	√	√	√	√	√	√
${\mathit{G}}_{\mathbf{3}}$	√	√	√	√	√	√
${\mathit{O}}_{\mathbf{4}}$	√	√	√	√	√	√
${\mathit{O}}_{\mathbf{5}}$	√	√	√	√	√	√

.

4. Order Preservation

In the following we show that the meet operation, join operation, convex combination, and ⊛-composition of overlap/grouping functions are order preserving.

Theorem 2.

Suppose that four overlap functions have $O_1⪯O_2$ and $O_3⪯O_4$, then $(O_1\vee O_3)⪯(O_2\vee O_4)$, $(O_1\wedge O_3)⪯(O_2\wedge O_4)$ $\left(O_{1,3,\lambda }\right)⪯\left(O_{2,4,\lambda }\right)$ and $(O_1⊛O_3)⪯(O_2⊛O_4)$, where $O_{1,3,\lambda }=\lambda O_1(\eta ,\xi )+(1-\lambda )O_3(\eta ,\xi )$ and $O_{2,4,\lambda }=\lambda O_2(\eta ,\xi )+(1-\lambda )O_4(\eta ,\xi )$.

Proof. 

The case for meet operation, join operation, and convex combination are straightforward. We show only that ⊛-composition preserves order. For any $\eta ,\xi \in [0,1]$, from $O_3⪯O_4$, we have $O_3(\eta ,\xi )\le O_4(\eta ,\xi )$. Since $O_1$ is non-decreasing and $O_1⪯O_2$, we have

$$\begin{array}{cc}(O_1⊛O_3)(\eta ,\xi )\hfill & =O_1\left(\eta ,O_3(\eta ,\xi )\right)\hfill \\ \hfill & \le O_1\left(\eta ,O_4(\eta ,\xi )\right)\hfill \\ \hfill & \le O_2\left(\eta ,O_4(\eta ,\xi )\right)\hfill \\ \hfill & =(O_2⊛O_4)(\eta ,\xi ).\hfill \end{array}$$

Thus, $(O_1⊛O_3)⪯(O_2⊛O_4)$. □

Theorem 3.

Suppose that four grouping functions have $G_1⪯G_2$ and $G_3⪯G_4$, then $(G_1\vee G_3)⪯(G_2\vee G_4)$, $(G_1\wedge G_3)⪯(G_2\wedge G_4)$ $\left(G_{1,3,\lambda }\right)⪯\left(G_{2,4,\lambda }\right)$ and $(G_1⊛G_3)⪯(G_2⊛G_4)$, where $G_{1,3,\lambda }=\lambda G_1(\eta ,\xi )+(1-\lambda )G_3(\eta ,\xi )$ and $G_{2,4,\lambda }=\lambda G_2(\eta ,\xi )+(1-\lambda )G_4(\eta ,\xi )$.

5. Properties Preservation

In the following, we study properties preserved by meet operation, join operation, convex combination, and ⊛-composition of overlap/grouping functions.

5.1. Properties Preserved by Meet and Join Operations of Overlap/Grouping Functions

First, we consider the meet and join operations of overlap/grouping functions.

Theorem 4.

If two overlap functions $O_1$ and $O_2$ satisfy (ID)((MI), (HO-k), (k-LI), (PS)), then $(O_1\vee O_2)$ and $(O_1\wedge O_2)$ also satisfy (ID)((MI), (HO-k), (k-LI), (PS)).

Proof. 

First, we show that meet operation preserves (ID). Assume that $O_1$ and $O_2$ satisfy (ID); then, for any $\lambda ,\eta \in [0,1]$,

$$\begin{array}{cc}(O_1\vee O_2)(\eta ,\eta )\hfill & =max\left(O_1(\eta ,\eta ),O_2(\eta ,\eta )\right)\hfill \\ \hfill & =max\left(\eta ,\eta \right)\hfill \\ \hfill & =\eta .\hfill \end{array}$$

Next, we show that meet operation preserves (MI). Assume that $O_1$ and $O_2$ satisfy (MI), then, for any $\alpha ,\eta ,\xi \in [0,1]$,

$$\begin{array}{cc}(O_1\vee O_2)(\alpha \eta ,\xi )\hfill & =max\left(O_1(\alpha \eta ,\xi ),O_2(\alpha \eta ,\xi )\right)\hfill \\ \hfill & =max\left(O_1(\eta ,\alpha \xi ),O_2(\eta ,\alpha \xi )\right)\hfill \\ \hfill & =(O_1\vee O_2)(\eta ,\alpha \xi ).\hfill \end{array}$$

Then, we show that the meet operation preserves (HO-k). Assuming that $O_1$ and $O_2$ satisfy (HO-k), then, for any $\alpha ,\eta ,\xi \in [0,1]$,

$$\begin{array}{cc}(O_1\vee O_2)(\alpha \eta ,\alpha \xi )\hfill & =max\left(O_1(\alpha \eta ,\alpha \xi ),O_2(\alpha \eta ,\alpha \xi )\right)\hfill \\ \hfill & =max\left({\alpha }^k{O}_1(\eta ,\xi ),{\alpha }^k{O}_2(\eta ,\xi )\right)\hfill \\ \hfill & ={\alpha }^{k}max\left(O_1(\eta ,\xi ),O_2(\eta ,\xi )\right)\hfill \\ \hfill & ={\alpha }^k(O_1\vee O_2)(\eta ,\xi ).\hfill \end{array}$$

Afterwards, we show that meet operation preserves (k-LI). Assume that $O_1$ and $O_2$ satisfy (k-LI), then, for any ${\eta }_1,{\eta }_2,{\xi }_1,{\xi }_2\in [0,1]$,

$$\begin{array}{cc}\hfill & |(O_1\vee O_2)({\eta }_1,{\xi }_1)-(O_1\vee O_2)({\eta }_2,{\xi }_2)|\hfill \\ \hfill & =|max\left(O_1({\eta }_1,{\xi }_1),O_2({\eta }_1,{\xi }_1)\right)-max\left(O_1({\eta }_2,{\xi }_2),O_2({\eta }_2,{\xi }_2)\right)|\hfill \\ \hfill & \le max\left(|O_1({\eta }_1,{\xi }_1)-O_1({\eta }_2,{\xi }_2)|,|O_2({\eta }_1,{\xi }_1)-O_2({\eta }_2,{\xi }_2)|\right)\hfill \\ \hfill & \le max\left(k\left(\right|{\eta }_1-{\eta }_2|+|{\xi }_1-{\xi }_2\left|\right),k\left(\right|{\eta }_1-{\eta }_2|+|{\xi }_1-{\xi }_2\left|\right)|\right)\hfill \\ \hfill & =k\left(\right|{\eta }_1-{\eta }_2|+|{\xi }_1-{\xi }_2\left|\right).\hfill \end{array}$$

Finally we show that meet operation preserves (PS). Assume that $O_1$ and $O_2$ satisfy (PS), then, for any $r,\eta ,\xi \in [0,1]$,

$$\begin{array}{cc}(O_1\vee O_2)({\eta }^r,{\xi }^r)\hfill & =max\left(O_1({\eta }^r,{\xi }^r),O_2({\eta }^r,{\xi }^r)\right)\hfill \\ \hfill & =max\left(O_1{(\eta ,\xi )}^r,O_2{(\eta ,\xi )}^r\right)\hfill \\ \hfill & ={\left(max\left(O_1(\eta ,\xi ),O_2(\eta ,\xi )\right)\right)}^r\hfill \\ \hfill & =(O_1\vee O_2){(\eta ,\xi )}^r.\hfill \end{array}$$

Similarly, we can show that the join operation also preserves (ID)((MI), (HO-k), (k-LI), (PS)). □

5.2. Properties Preserved by Convex Combination of Overlap/Grouping Functions

Second, we consider the convex combination of overlap/grouping functions.

Theorem 5.

If two overlap functions $O_1$ and $O_2$ satisfy (ID)((MI), (HO-k), (k-LI)), then, for any $\lambda \in [0,1]$, their convex combination of $O_{\lambda }$ also satisfies (ID)((MI), (HO-k), (k-LI)).

Proof. 

First, we show that convex combination preserves (ID). Assume that $O_1$ and $O_2$ satisfy (ID), then, for any $\lambda ,\eta \in [0,1]$,

$$\begin{array}{cc}{O}_{\lambda }(\eta ,\eta )\hfill & =\lambda O_1(\eta ,\eta )+(1-\lambda )O_2(\eta ,\eta )\hfill \\ \hfill & =\lambda \eta +(1-\lambda )\eta \hfill \\ \hfill & =\eta .\hfill \end{array}$$

Next, we show that convex combination preserves (MI). Assume that $O_1$ and $O_2$ satisfy (MI), then, for any $\lambda ,\alpha ,\eta ,\xi \in [0,1]$,

$$\begin{array}{cc}{O}_{\lambda }(\alpha \eta ,\xi )\hfill & =\lambda O_1(\alpha \eta ,\xi )+(1-\lambda )O_2(\alpha \eta ,\xi )\hfill \\ \hfill & =\lambda O_1(\eta ,\alpha \xi )+(1-\lambda )O_2(\eta ,\alpha \xi )\hfill \\ \hfill & =O_{\lambda }(\eta ,\alpha \xi ).\hfill \end{array}$$

Then, we show that convex combination preserves (HO-k). Assume that $O_1$ and $O_2$ satisfy (HO-k), then, for any $\lambda ,\alpha ,\eta ,\xi \in [0,1]$,

$$\begin{array}{cc}{O}_{\lambda }(\alpha \eta ,\alpha \xi )\hfill & =\lambda O_1(\alpha \eta ,\alpha \xi )+(1-\lambda )O_2(\alpha \eta ,\alpha \xi )\hfill \\ \hfill & =\lambda {\alpha }^k{O}_1(\eta ,\xi )+(1-\lambda ){\alpha }^k{O}_2(\eta ,\xi )\hfill \\ \hfill & ={\alpha }^k\left(\lambda O_1(\eta ,\xi )+(1-\lambda )O_2(\eta ,\xi )\right)\hfill \\ \hfill & ={\alpha }^k{O}_{\lambda }(\eta ,\xi ).\hfill \end{array}$$

Finally, we show that convex combination preserves (k-LI). Assume that $O_1$ and $O_2$ satisfy (k-LI), then, for any $\lambda ,\alpha ,\eta ,\xi \in [0,1]$,

$$\begin{array}{cc}\hfill & |O_{\lambda }({\eta }_1,{\xi }_1)-O_{\lambda }({\eta }_2,{\xi }_2)|\hfill \\ \hfill & =|\lambda O_1({\eta }_1,{\xi }_1)+(1-\lambda )O_2({\eta }_1,{\xi }_1)-\lambda O_1({\eta }_2,{\xi }_2)-(1-\lambda )O_2({\eta }_2,{\xi }_2)|\hfill \\ \hfill & =|\lambda \left(O_1({\eta }_1,{\xi }_1)-O_1({\eta }_2,{\xi }_2)\right)+(1-\lambda )\left(O_2({\eta }_1,{\xi }_1)-O_2({\eta }_2,{\xi }_2)\right)|\hfill \\ \hfill & \le \left|\lambda k\right(|{\eta }_1-{\eta }_2|+|{\xi }_1-{\xi }_2\left|\right)+(1-\lambda )k\left(\right|{\eta }_1-{\eta }_2|+|{\xi }_1-{\xi }_2\left|\right)|\hfill \\ \hfill & =k\left(\right|{\eta }_1-{\eta }_2|+|{\xi }_1-{\xi }_2\left|\right).\hfill \end{array}$$

□

Note that convex combination does not preserve (PS), since we have

$$\begin{array}{cc}{O}_{\lambda }({\eta }^r,{\xi }^r)\hfill & =\lambda O_1({\eta }^r,{\xi }^r)+(1-\lambda )O_2({\eta }^r,{\xi }^r)\hfill \\ \hfill & =\lambda O_1{(\eta ,\xi )}^r+(1-\lambda )O_2{(\eta ,\xi )}^r,\hfill \end{array}$$

and

$$\begin{array}{cc}{O}_{\lambda }{(\eta ,\xi )}^r\hfill & ={\left(\lambda O_1(\eta ,\xi )+(1-\lambda )O_2(\eta ,\xi )\right)}^r\hfill \\ \hfill & \ne \lambda O_1{(\eta ,\xi )}^r+(1-\lambda )O_2{(\eta ,\xi )}^r\hfill \end{array}$$

for some $\lambda ,r,\eta ,\xi \in [0,1]$.

5.3. Properties Preserved by ⊛-Composition of Overlap/Grouping Functions

Third, we consider the ⊛-composition of overlap/grouping functions.

Theorem 6.

If two overlap functions $O_1$ and $O_2$ satisfy (ID)((HO-1), (PS)), then, their ⊛-composition $(O_1⊛O_2)$ also satisfies (ID)((HO-1), (PS)).

Proof. 

First, we show that ⊛-composition preserves (ID). Assume that $O_1$ and $O_2$ satisfy (ID), then, for any $\lambda ,\eta \in [0,1]$,

$$\begin{array}{cc}(O_1⊛O_2)(\eta ,\eta )\hfill & =O_1\left(\eta ,O_2(\eta ,\eta )\right)\hfill \\ \hfill & =O_1\left(\eta ,\eta \right)\hfill \\ \hfill & =\eta .\hfill \end{array}$$

Next, we show that ⊛-composition preserves (HO-1). Assume that $O_1$ and $O_2$ satisfy (HO-1), then, for any $\alpha ,\eta ,\xi \in [0,1]$,

$$\begin{array}{cc}(O_1⊛O_2)(\alpha \eta ,\alpha \xi )\hfill & =O_1\left(\alpha \eta ,O_2(\alpha \eta ,\alpha \xi )\right)\hfill \\ \hfill & =O_1\left(\alpha \eta ,\alpha O_2(\eta ,\xi )\right)\hfill \\ \hfill & =\alpha O_1\left(\eta ,O_2(\eta ,\xi )\right)\hfill \\ \hfill & =\alpha (O_1⊛O_2)(\eta ,\xi ).\hfill \end{array}$$

Then, we show that ⊛-composition preserves (PS). Assume that $O_1$ and $O_2$ satisfy (PS), then, for any $r,\eta ,\xi \in [0,1]$,

$$\begin{array}{cc}(O_1⊛O_2)({\eta }^r,{\xi }^r)\hfill & =O_1\left({\eta }^r,O_2({\eta }^r,{\xi }^r)\right)\hfill \\ \hfill & =O_1\left({\eta }^r,O_2{(\eta ,\xi )}^r\right)\hfill \\ \hfill & =O_1{\left(\eta ,O_2(\eta ,\xi )\right)}^r\hfill \\ \hfill & =(O_1⊛O_2){(\eta ,\xi )}^r.\hfill \end{array}$$

□

Note that we only show that ⊛-composition preserves (HO-1), it does not preserve (HO-k) for $k\in ]0,\infty [$ and $k\ne 1$. For example, let $O_1(\eta ,\xi )=O_2(\eta ,\xi )={\eta }^2{\xi }^2$, then $(O_1⊛O_2)(\eta ,\xi )={\eta }^6{\xi }^4$, we know that $O_1$ and $O_2$ satisfy (HO-2), i.e., $O_1(\alpha \eta ,\alpha \xi )={\alpha }^2{O}_1(\eta ,\xi )$, but $(O_1⊛O_2)(\eta ,\xi )$ does not satisfy (HO-2) since $(O_1⊛O_2)(\alpha \eta ,\alpha \xi )={\alpha }^{10}{\eta }^6{\xi }^4\ne {\alpha }^2{\eta }^6{\xi }^4={\alpha }^2(O_1⊛O_2)(\eta ,\xi )$.

The ⊛-composition does not preserve (MI). Assume that $O_1$ and $O_2$ satisfy (MI), then

$$\begin{array}{cc}(O_1⊛O_2)(\eta ,\alpha \xi )\hfill & =O_1\left(\eta ,O_2(\eta ,\alpha \xi )\right)\hfill \\ \hfill & =O_1\left(\eta ,O_2(\alpha \eta ,\xi )\right)\hfill \\ \hfill & \ne O_1\left(\alpha \eta ,O_2(\alpha \eta ,\xi )\right)\hfill \\ \hfill & =(O_1⊛O_2)(\alpha \eta ,\xi )\hfill \end{array}$$

for some $\alpha ,\eta ,\xi \in [0,1]$.

The ⊛-composition does not preserve (k-LI).

Example 1.

Let $O_1(\eta ,\xi )=O_2(\eta ,\xi )=\eta \xi$, then $(O_1⊛O_2)(\eta ,\xi )={\eta }^2\xi$,

$$\begin{array}{cc}|O_1({\eta }_1,{\xi }_1)-O_2({\eta }_2,{\xi }_2)|\hfill & = |{\eta }_1{\xi }_1-{\eta }_2{\xi }_2|\hfill \\ \hfill & = |{\eta }_1{\xi }_1-{\eta }_1{\xi }_2+{\eta }_1{\xi }_2-{\eta }_2{\xi }_2|\hfill \\ \hfill & = |{\eta }_1({\xi }_1-{\xi }_2)+{\xi }_2({\eta }_1-{\eta }_2)|\hfill \\ \hfill & \le |{\eta }_1({\xi }_1-{\xi }_2)|+|{\xi }_2({\eta }_1-{\eta }_2)|\hfill \\ \hfill & \le |{\xi }_1-{\xi }_2|+|{\eta }_1-{\eta }_2|.\hfill \end{array}$$

Thus, $O_1$ and $O_2$ satisfy (1-LI). Let ${\eta }_1={\xi }_1=0.8$ and ${\eta }_2={\xi }_2=1,$ then $(O_1⊛O_2)(0.8,0.8)-(O_1⊛O_2)(1,1)=0.488>0.4=\left(\right|0.8-1|+|0.8-1\left|\right)$, so $O_1⊛O_2$ does not satisfy (1-LI).

However, we have the following result.

Theorem 7.

If two overlap functions $O_1$ and $O_2$ respectively satisfy (k₁-LI) and (k₂-LI), then their ⊛-composition $(O_1⊛O_2)$ satisfies ((k₁+ k₁k₂)-LI).

Proof. 

Assume that $O_1$ and $O_2$ respectively satisfy (k₁-LI) and (k₂-LI), then, for any ${\eta }_1,{\eta }_2,{\xi }_1,{\xi }_2\in [0,1]$, we have

$$\begin{array}{cc}|(O_1⊛O_2)({\eta }_1,{\xi }_1)-(O_1⊛O_2)({\eta }_2,{\xi }_2)|\hfill & =|O_1({\eta }_1,O_2({\eta }_1,{\xi }_1))-O_1({\eta }_2,O_2({\eta }_2,{\xi }_2))|\hfill \\ \hfill & \le k_1\left(|{\eta }_1-{\eta }_2|+|O_2({\eta }_1,{\xi }_1)-O_2({\eta }_2,{\xi }_2)|\right)\hfill \\ \hfill & \le k_1\left(|{\eta }_1-{\eta }_2|+k_2|{\eta }_1-{\eta }_2|+k_2|{\xi }_1-{\xi }_2|\right)\hfill \\ \hfill & =(k_1+k_1{k}_2)|{\eta }_1-{\eta }_2|+k_1{k}_2|{\xi }_1-{\xi }_2|\hfill \\ \hfill & \le (k_1+k_1{k}_2)\left(|{\eta }_1-{\eta }_2|+|{\xi }_1-{\xi }_2|\right).\hfill \end{array}$$

□

5.4. Summary

Thus far, we have studied the basic properties of overlap/grouping functions w.r.t. the meet operation, join operation, convex combination, and ⊛-composition. The summary of the properties of overlap/grouping functions w.r.t. the meet operation, join operation, convex combination, and ⊛-composition is shown in

Table 2. Properties preservation of the compositions.

Property	${\mathit{O}}_1$	${\mathit{O}}_2$	${\mathit{O}}_1\vee {\mathit{O}}_2$	${\mathit{O}}_1\wedge {\mathit{O}}_2$	${\mathit{O}}_{\mathit{\lambda }}$	${\mathit{O}}_1⊛{\mathit{O}}_2$
ID	√	√	√	√	√	√
MI	√	√	√	√	√	×
HO-k	√	√	√	√	√	×
k-LI	√	√	√	√	√	×
PS	√	√	√	√	×	√

.

6. Conclusions

This paper studies the properties preservation of overlap/grouping functions w.r.t. meet operation, join operation, convex combination, and ⊛-composition. The main conclusions are listed as follows.

(1)

Closures of two bivariate functions w.r.t. meet operation, join operation, convex combination, and ⊛-composition have been obtained in Table 1. Note that ⊛-composition does not preserve (O1), and ⊛-composition of overlap/grouping functions is not closed. In other words, ⊛-composition can not be used to generate new overlap/grouping functions.

(2)

We show that meet operation, join operation, convex combination, and ⊛-composition of overlap/grouping functions are order preserving, see Theorems 2 and 3.

(3)

We have investigated the preservation of the law of (ID), (MI), (HO-k), (k-LI), and (PS) w.r.t. meet operation, join operation, convex combination, and ⊛-composition, which can be summarized in Table 2.

These results can be served as a certain criteria for choices of generation methods of overlap/grouping functions from given overlap/grouping functions. For example, convex combination does not preserve (PS). Thus, we can not generate a power stable overlap function from two power stable overlap functions by their convex combination.

As we know, overlap/grouping functions have been extended to interval-valued and complex-valued overlap/grouping functions. Could similar results be carried over to the interval-valued and complex-valued settings? Moreover, special overlap/grouping functions such as Archimedean and multiplicatively generated overlap/grouping functions have been studied. In these cases, many restrictions have been added. For further works, it follows that we intend to consider properties preservation of these overlap/grouping functions w.r.t. different composition methods.