*Article* **On the Composition of Overlap and Grouping Functions**

**Songsong Dai , Lei Du, Haifeng Song and Yingying Xu \***

School of Electronics and Information Engineering, Taizhou University, Taizhou 318000, China; ssdai@tzc.edu.cn (S.D.); dulei2109@tzc.edu.cn (L.D.); isshf@126.com (H.S.)

**\*** Correspondence: yyxu@tzc.edu.cn

**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.

**Keywords:** overlap functions; grouping functions; composition; closures; properties preservation

### **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 [9–17]. 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 [9,12,13,18–21]. 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 [22–27].

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

**Citation:** Dai, S.; Du, L.; Song, H.; Xu, Y. On the Composition of Overlap and Grouping Functions. *Axioms* **2021**, *10*, 272. https://doi.org/ 10.3390/axioms10040272

Academic Editor: Amit K. Shukla

Received: 9 September 2021 Accepted: 20 October 2021 Published: 24 October 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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] <sup>2</sup> <sup>→</sup> [0, 1] *is an overlap function if it has the following properties:*


**Definition 2** ([1])**.** *A bivariate function G* : [0, 1] <sup>2</sup> <sup>→</sup> [0, 1] *is a grouping function if it has the following properties:*


If *O* is an overlap function, then the function *G*(*η*, *ξ*) = 1 − *O*(1 − *η*, 1 − *ξ*) 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*0 , if *O*(*η*, *ξ*) ≤ *O*<sup>0</sup> (*η*, *ξ*) holds for all (*η*, *ξ*) ∈ [0, 1] 2 , then we say that *O* is weaker than *O*0 , denoted *O O*<sup>0</sup> . For example, consider the following three overlap functions *OM*(*η*, *ξ*) = min(*η*, *ξ*), *OP*(*η*, *ξ*) = *ηξ* and *<sup>O</sup>Mid*(*η*, *<sup>ξ</sup>*) = *ηξ <sup>η</sup>*+*<sup>ξ</sup>* 2 , we get this ordering for these overlap functions:

$$O\_{Mid} \preceq O\_P \preceq O\_{M\cdot}$$

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

**(ID)** Idempotency:

*O*(*η*, *η*) = *η*

for all *η* ∈ [0, 1];

**(MI)** Migrativity:

*O*(*αη*, *ξ*) = *O*(*η*, *αξ*)

for all *α*, *η*, *ξ* ∈ [0, 1];

**(HO-***k***)** Homogeneous of order *k* ∈]0, ∞[:

*O*(*αη*, *αξ*) = *α <sup>k</sup>O*(*η*, *ξ*)

for all *α* ∈ [0, ∞[ and *η*, *ξ* ∈ [0, 1] such that *αη*, *αξ* ∈ [0, 1];

**(***k***-LI)** k-Lipschitz:

$$|O(\eta\_1, \mathfrak{f}\_1) - O(\eta\_2, \mathfrak{f}\_2)| \le k(|\eta\_1 - \eta\_2| + |\mathfrak{f}\_1 - \mathfrak{f}\_2|)$$

for all *η*1, *η*2, *ξ*1, *ξ*<sup>2</sup> ∈ [0, 1].

**(PS)** Power stable [29]:

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

for all *r* ∈]0, ∞[ and *η*, *ξ* ∈ [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*<sup>1</sup> and *O*2, meet and join operations of *O*<sup>1</sup> and *O*<sup>2</sup> are defined by

$$(O\_1 \lor O\_2)(\eta, \xi) = \max\left(O\_1(\eta, \xi), O\_2(\eta, \xi)\right),\tag{1}$$

$$(O\_1 \wedge O\_2)(\eta, \xi) = \min\left(O\_1(\eta, \xi), O\_2(\eta, \xi)\right) \tag{2}$$

for all (*η*, *ξ*) ∈ [0, 1] 2 .

For any two overlap (or grouping) functions *O*<sup>1</sup> and *O*2, a convex combination of *O*<sup>1</sup> and *O*<sup>2</sup> is defined as

$$O\_{\lambda} = \lambda O\_1(\eta, \xi) + (1 - \lambda) O\_2(\eta, \xi) \tag{3}$$

for all (*η*, *ξ*) ∈ [0, 1] <sup>2</sup> and *<sup>λ</sup>* <sup>∈</sup> [0, 1].

For any two overlap (or grouping) functions *O*<sup>1</sup> and *O*2, the ~-composition of *O*<sup>1</sup> and *O*<sup>2</sup> is defined as

$$(O\_1 \circledast O\_2)(\eta, \xi) = O\_1(\eta, O\_2(\eta, \xi))\tag{4}$$

for all (*η*, *ξ*) ∈ [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 (*O***1**), (*O***2**), (*O***3**), (*O***4**) and (*O***5**). Similarly, the ~-composition of two grouping functions is closed means ~-composition of two bivariate functions on [0, 1] preserves (*G***1**), (*G***2**), (*G***3**), (*G***4**) and (*G***5**).

**Theorem 1.** *If two bivariate functions O*1,*O*<sup>2</sup> : [0, 1] <sup>2</sup> <sup>→</sup> [0, 1] *satisfy (O***2***) (O***3***), (G***2***), (G***3***), (O***4***), (O***5***) , then* (*O*<sup>1</sup> ~ *O*2) *also satisfies (O***2***) (O***3***), (G***2***), (G***3***), (O***4***), (O***5***) .*

**Proof.** First, we show that ~-composition preserves (*O***2**). If

$$(O\_1 \circledast O\_2)(\eta, \xi) = O\_1(\eta, O\_2(\eta, \xi)) = 0,$$

then, since *O*<sup>1</sup> satisfies (*O***2**), we have *ηO*2(*η*, *ξ*)=0. Case I, if *η* = 0 and *O*2(*η*, *ξ*) 6= 0, then *ηξ* = 0*ξ* = 0; Case II, if *η* = 0 and *O*2(*η*, *ξ*) = 0, then *ηξ* = 0*ξ* = 0; Case III, if *η* 6= 0 and *O*2(*η*, *ξ*) = 0, since *O*<sup>2</sup> satisfies (*O***2**), then *ηξ* = 0.

Next, we show that ~-composition preserves (*O***3**). If

$$(\mathcal{O}\_1 \circledast \mathcal{O}\_2)(\eta\_\prime \mathfrak{f}) = \mathcal{O}\_1(\eta\_\prime \mathcal{O}\_2(\eta\_\prime \mathfrak{f})) = 1,$$

then, since *O*<sup>1</sup> satisfies (*O***3**), we have *ηO*2(*η*, *ξ*)=1. Then, *η* = 1 and *O*2(*η*, *ξ*) = 1, since *O*<sup>2</sup> satisfies (*O***3**), then *ηξ* = 1.

Then, we show that ~-composition preserves (*G***2**). If

$$(\mathcal{O}\_1 \circledast \mathcal{O}\_2)(\eta\_\prime \mathfrak{F}) = \mathcal{O}\_1(\eta\_\prime \mathcal{O}\_2(\eta\_\prime \mathfrak{F})) = \mathfrak{0}\_\prime$$

then, since *O*<sup>1</sup> satisfies (*G***2**), we have *η* = *O*2(*η*, *ξ*)=0. Since *O*<sup>2</sup> satisfies (*G***2**), then *η* = *ξ* = 0.

Afterwards, we show that ~-composition preserves (*G***3**). If

$$(\mathcal{O}\_1 \circledast \mathcal{O}\_2)(\eta, \xi) = \mathcal{O}\_1(\eta, \mathcal{O}\_2(\eta, \xi)) = 1, 2$$

then, since *O*<sup>1</sup> satisfies (*G***3**), we have *η* = 1 or *O*2(*η*, *ξ*)=1. Since *O*<sup>2</sup> satisfies (*G***3**), *O*2(*η*, *ξ*)=1 means *η* = 1 or *ξ* = 1.

The case for (*O***4**) and (*O***5**) are straightforward.

Unfortunately, ~-composition of two bivariate functions does not preserve (*O***1**). For example, let *O*1(*η*, *ξ*) = *O*2(*η*, *ξ*) = *ηξ*; then, (*O*<sup>1</sup> ~ *O*2)(*η*, *ξ*) = *η* 2 *ξ* 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(*η*, *ξ*) = *O*2(*η*, *ξ*) = min(*η*, *ξ*), their ~-composition (*O*<sup>1</sup> ~ *O*2)(*η*, *ξ*) = min(*η*, *ξ*) is an overlap function.

The summary of the closures of two bivariate functions w.r.t. these compositions is shown in Table 1.


**Table 1.** Closures of the compositions.

### **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*<sup>1</sup> *O*<sup>2</sup> *and O*<sup>3</sup> *O*4*, then* (*O*<sup>1</sup> ∨ *O*3) (*O*<sup>2</sup> ∨ *O*4)*,* (*O*<sup>1</sup> ∧ *O*3) (*O*<sup>2</sup> ∧ *O*4) (*O*1,3,*λ*) (*O*2,4,*λ*) *and* (*O*<sup>1</sup> ~ *O*3) (*O*<sup>2</sup> ~ *O*4)*, where O*1,3,*<sup>λ</sup>* = *λO*1(*η*, *ξ*) + (1 − *λ*)*O*3(*η*, *ξ*) *and O*2,4,*<sup>λ</sup>* = *λO*2(*η*, *ξ*) + (1 − *λ*)*O*4(*η*, *ξ*)*.*

**Proof.** The case for meet operation, join operation, and convex combination are straightforward. We show only that ~-composition preserves order. For any *η*, *ξ* ∈ [0, 1], from *O*<sup>3</sup> *O*4, we have *O*3(*η*, *ξ*) ≤ *O*4(*η*, *ξ*). Since *O*<sup>1</sup> is non-decreasing and *O*<sup>1</sup> *O*2, we have

$$\begin{array}{rcl} (O\_1 \circledast O\_3)(\eta, \mathfrak{f}) &= O\_1(\eta, O\_3(\eta, \mathfrak{f})) \\ &\leq O\_1(\eta, O\_4(\eta, \mathfrak{f})) \\ &\leq O\_2(\eta, O\_4(\eta, \mathfrak{f})) \\ &= (O\_2 \circledast O\_4)(\eta, \mathfrak{f}). \end{array}$$

Thus, (*O*<sup>1</sup> ~ *O*3) (*O*<sup>2</sup> ~ *O*4).

**Theorem 3.** *Suppose that four grouping functions have G*<sup>1</sup> *G*<sup>2</sup> *and G*<sup>3</sup> *G*4*, then* (*G*<sup>1</sup> ∨ *G*3) (*G*<sup>2</sup> ∨ *G*4)*,* (*G*<sup>1</sup> ∧ *G*3) (*G*<sup>2</sup> ∧ *G*4) (*G*1,3,*λ*) (*G*2,4,*λ*) *and* (*G*<sup>1</sup> ~ *G*3) (*G*<sup>2</sup> ~ *G*4)*, where G*1,3,*<sup>λ</sup>* = *λG*1(*η*, *ξ*) + (1 − *λ*)*G*3(*η*, *ξ*) *and G*2,4,*<sup>λ</sup>* = *λG*2(*η*, *ξ*) + (1 − *λ*)*G*4(*η*, *ξ*)*.*

### **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*<sup>1</sup> *and O*<sup>2</sup> *satisfy (***ID***) (***MI***), (***HO-***k), (k***-LI***), (***PS***) , then* (*O*<sup>1</sup> ∨ *O*2) *and* (*O*<sup>1</sup> ∧ *O*2) *also satisfy (***ID***) (***MI***), (***HO-***k), (k***-LI***), (***PS***) .*

**Proof.** First, we show that meet operation preserves (**ID**). Assume that *O*<sup>1</sup> and *O*<sup>2</sup> satisfy (**ID**); then, for any *λ*, *η* ∈ [0, 1],

$$\begin{array}{rcl} (O\_1 \lor O\_2)(\eta, \eta) &= \max \left( O\_1(\eta, \eta), O\_2(\eta, \eta) \right) \\ &= \max \left( \eta, \eta \right) \\ &= \eta. \end{array}$$

Next, we show that meet operation preserves (**MI**). Assume that *O*<sup>1</sup> and *O*<sup>2</sup> satisfy (**MI**), then, for any *α*, *η*, *ξ* ∈ [0, 1],

$$\begin{array}{rcl} (O\_1 \lor O\_2)(\alpha \eta, \xi) &= \max\left(O\_1(\alpha \eta, \xi), O\_2(\alpha \eta, \xi)\right) \\ &= \max\left(O\_1(\eta, \alpha \xi), O\_2(\eta, \alpha \xi)\right) \\ &= (O\_1 \lor O\_2)(\eta, \alpha \xi). \end{array}$$

Then, we show that the meet operation preserves (**HO-***k*). Assuming that *O*<sup>1</sup> and *O*<sup>2</sup> satisfy (**HO-***k*), then, for any *α*, *η*, *ξ* ∈ [0, 1],

$$\begin{array}{rcl} (O\_1 \lor O\_2)(\mathfrak{a}\eta, \mathfrak{a}\mathfrak{f}) &=& \max\left(O\_1(\mathfrak{a}\eta, \mathfrak{a}\mathfrak{f}), O\_2(\mathfrak{a}\eta, \mathfrak{a}\mathfrak{f})\right) \\ &=& \max\left(\mathfrak{a}^k O\_1(\mathfrak{\eta}, \mathfrak{f}), \mathfrak{a}^k O\_2(\mathfrak{\eta}, \mathfrak{f})\right) \\ &=& \mathfrak{a}^k \max\left(O\_1(\mathfrak{\eta}, \mathfrak{f}), O\_2(\mathfrak{\eta}, \mathfrak{f})\right) \\ &=& \mathfrak{a}^k (O\_1 \lor O\_2)(\mathfrak{\eta}, \mathfrak{f}). \end{array}$$

Afterwards, we show that meet operation preserves (*k***-LI**). Assume that *O*<sup>1</sup> and *O*<sup>2</sup> satisfy (*k***-LI**), then, for any *η*1, *η*2, *ξ*1, *ξ*<sup>2</sup> ∈ [0, 1],

> |(*O*<sup>1</sup> ∨ *O*2)(*η*1, *ξ*1) − (*O*<sup>1</sup> ∨ *O*2)(*η*2, *ξ*2)| =  max *O*1(*η*1, *ξ*1),*O*2(*η*1, *ξ*1) <sup>−</sup> max *O*1(*η*2, *ξ*2),*O*2(*η*2, *ξ*2) <sup>≤</sup> max  *O*1(*η*1, *ξ*1) − *O*1(*η*2, *ξ*2) , *O*2(*η*1, *ξ*1) − *O*2(*η*2, *ξ*2) <sup>≤</sup> max *k*(|*η*<sup>1</sup> − *η*2| + |*ξ*<sup>1</sup> − *ξ*2|), *k*(|*η*<sup>1</sup> − *η*2| + |*ξ*<sup>1</sup> − *ξ*2|) = *k*(|*η*<sup>1</sup> − *η*2| + |*ξ*<sup>1</sup> − *ξ*2|).

Finally we show that meet operation preserves (**PS**). Assume that *O*<sup>1</sup> and *O*<sup>2</sup> satisfy (**PS**), then, for any *r*, *η*, *ξ* ∈ [0, 1],

$$\begin{array}{rcl} (O\_1 \vee O\_2)(\eta^r, \xi^r) &= \max\left(O\_1(\eta^r, \xi^r), O\_2(\eta^r, \xi^r)\right) \\ &= \max\left(O\_1(\eta, \xi)^r, O\_2(\eta, \xi)^r\right) \\ &= \left(\max\left(O\_1(\eta, \xi), O\_2(\eta, \xi)\right)\right)^r \\ &= (O\_1 \vee O\_2)(\eta, \xi)^r. \end{array}$$

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*<sup>1</sup> *and O*<sup>2</sup> *satisfy (***ID***) (***MI***), (***HO-***k), (k***-LI***) , then, for any λ* ∈ [0, 1]*, their convex combination of O<sup>λ</sup> also satisfies (***ID***) (***MI***), (***HO-***k), (k***-LI***) .*

**Proof.** First, we show that convex combination preserves (**ID**). Assume that *O*<sup>1</sup> and *O*<sup>2</sup> satisfy (**ID**), then, for any *λ*, *η* ∈ [0, 1],

$$\begin{array}{rcl} O\_{\lambda}(\eta,\eta) &= \lambda O\_{1}(\eta,\eta) + (1-\lambda)O\_{2}(\eta,\eta) \\ &= \lambda \eta + (1-\lambda)\eta \\ &= \eta. \end{array}$$

Next, we show that convex combination preserves (**MI**). Assume that *O*<sup>1</sup> and *O*<sup>2</sup> satisfy (**MI**), then, for any *λ*, *α*, *η*, *ξ* ∈ [0, 1],

$$\begin{array}{rcl} O\_{\lambda}(\mathfrak{a}\eta, \mathfrak{f}) &= \lambda O\_{1}(\mathfrak{a}\eta, \mathfrak{f}) + (1 - \lambda) O\_{2}(\mathfrak{a}\eta, \mathfrak{f}) \\ &= \lambda O\_{1}(\mathfrak{\eta}, \mathfrak{a}\mathfrak{f}) + (1 - \lambda) O\_{2}(\mathfrak{\eta}, \mathfrak{a}\mathfrak{f}) \\ &= O\_{\lambda}(\mathfrak{\eta}, \mathfrak{a}\mathfrak{f}). \end{array}$$

Then, we show that convex combination preserves (**HO-***k*). Assume that *O*<sup>1</sup> and *O*<sup>2</sup> satisfy (**HO-***k*), then, for any *λ*, *α*, *η*, *ξ* ∈ [0, 1],

$$\begin{array}{rcl} O\_{\lambda}(\mathfrak{a}\eta, \mathfrak{a}\mathfrak{f}) &=& \lambda O\_{1}(\mathfrak{a}\eta, \mathfrak{a}\mathfrak{f}) + (1 - \lambda) O\_{2}(\mathfrak{a}\eta, \mathfrak{a}\mathfrak{f}) \\ &=& \lambda \mathfrak{a}^{k} O\_{1}(\mathfrak{\eta}, \mathfrak{f}) + (1 - \lambda) \mathfrak{a}^{k} O\_{2}(\mathfrak{\eta}, \mathfrak{f}) \\ &=& \mathfrak{a}^{k}(\lambda O\_{1}(\mathfrak{\eta}, \mathfrak{f}) + (1 - \lambda) O\_{2}(\mathfrak{\eta}, \mathfrak{f})) \\ &=& \mathfrak{a}^{k} O\_{\lambda}(\mathfrak{\eta}, \mathfrak{f}). \end{array}$$

Finally, we show that convex combination preserves (*k***-LI**). Assume that *O*<sup>1</sup> and *O*<sup>2</sup> satisfy (*k***-LI**), then, for any *λ*, *α*, *η*, *ξ* ∈ [0, 1],

$$\begin{array}{l} \left| O\_{\lambda}(\eta\_{1},\xi\_{1}) - O\_{\lambda}(\eta\_{2},\xi\_{2}) \right| \\ = \left| \lambda O\_{1}(\eta\_{1},\xi\_{1}) + (1-\lambda)O\_{2}(\eta\_{1},\xi\_{1}) - \lambda O\_{1}(\eta\_{2},\xi\_{2}) - (1-\lambda)O\_{2}(\eta\_{2},\xi\_{2}) \right| \\ = \left| \lambda \left( O\_{1}(\eta\_{1},\xi\_{1}) - O\_{1}(\eta\_{2},\xi\_{2}) \right) + (1-\lambda) \left( O\_{2}(\eta\_{1},\xi\_{1}) - O\_{2}(\eta\_{2},\xi\_{2}) \right) \right| \\ \leq \left| \lambda k(|\eta\_{1} - \eta\_{2}| + |\tilde{\xi}\_{1} - \tilde{\xi}\_{2}|) + (1-\lambda)k(|\eta\_{1} - \eta\_{2}| + |\tilde{\xi}\_{1} - \tilde{\xi}\_{2}|) \right| \\ = k(|\eta\_{1} - \eta\_{2}| + |\tilde{\xi}\_{1} - \tilde{\xi}\_{2}|). \end{array}$$

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

$$\begin{aligned} \mathcal{O}\_{\lambda}(\eta^r, \mathfrak{F}^r) &= \lambda \mathcal{O}\_1(\eta^r, \mathfrak{F}^r) + (1 - \lambda) \mathcal{O}\_2(\eta^r, \mathfrak{F}^r) \\ &= \lambda \mathcal{O}\_1(\eta, \mathfrak{F})^r + (1 - \lambda) \mathcal{O}\_2(\eta, \mathfrak{F})^r \end{aligned}$$

and

$$\begin{aligned} O\_{\lambda}(\eta,\mathfrak{f})^{r} &= \begin{pmatrix} \lambda O\_{1}(\eta,\mathfrak{f}) + (1-\lambda)O\_{2}(\eta,\mathfrak{f}) \\ \neq \lambda O\_{1}(\eta,\mathfrak{f})^{r} + (1-\lambda)O\_{2}(\eta,\mathfrak{f})^{r} \end{pmatrix} \end{aligned}$$

for some *λ*,*r*, *η*, *ξ* ∈ [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*<sup>1</sup> *and O*<sup>2</sup> *satisfy (***ID***) (***HO-1***), (***PS***) , then, their* ~ *composition* (*O*<sup>1</sup> ~ *O*2) *also satisfies (***ID***) (***HO-1***), (***PS***) .*

**Proof.** First, we show that ~-composition preserves (**ID**). Assume that *O*<sup>1</sup> and *O*<sup>2</sup> satisfy (**ID**), then, for any *λ*, *η* ∈ [0, 1],

$$\begin{array}{rcl} (\mathcal{O}\_1 \circledast \mathcal{O}\_2)(\eta, \eta) &= \mathcal{O}\_1(\eta, \mathcal{O}\_2(\eta, \eta)) \\ &= \mathcal{O}\_1(\eta, \eta) \\ &= \eta. \end{array}$$

Next, we show that ~-composition preserves (**HO-1**). Assume that *O*<sup>1</sup> and *O*<sup>2</sup> satisfy (**HO-1**), then, for any *α*, *η*, *ξ* ∈ [0, 1],

$$\begin{array}{rcl} (O\_1 \circledast O\_2)(\alpha \eta \,\, \alpha \xi) &=& O\_1(\alpha \eta \,\, \alpha O\_2(\alpha \eta \,\, \alpha \xi)) \\ &=& O\_1(\alpha \eta \,\, \alpha O\_2(\eta \,\, \xi)) \\ &=& \alpha O\_1(\eta \,\, \alpha O\_2(\eta \,\, \xi)) \\ &=& \alpha (O\_1 \circledast O\_2)(\eta \,\, \xi). \end{array}$$

Then, we show that ~-composition preserves (**PS**). Assume that *O*<sup>1</sup> and *O*<sup>2</sup> satisfy (**PS**), then, for any *r*, *η*, *ξ* ∈ [0, 1],

$$\begin{array}{rcl} (\mathcal{O}\_1 \circledast \mathcal{O}\_2)(\eta^r, \mathfrak{z}^r) &= \mathcal{O}\_1 \left( \eta^r, \mathcal{O}\_2(\eta^r, \mathfrak{z}^r) \right) \\ &= \mathcal{O}\_1 \left( \eta^r, \mathcal{O}\_2(\eta, \mathfrak{z})^r \right) \\ &= \mathcal{O}\_1 \left( \eta, \mathcal{O}\_2(\eta, \mathfrak{z}) \right)^r \\ &= \left( \mathcal{O}\_1 \circledast \mathcal{O}\_2 \right) (\eta, \mathfrak{z})^r. \end{array}$$

Note that we only show that ~-composition preserves (**HO-1**), it does not preserve (**HO-***k*) for *k* ∈]0, ∞[ and *k* 6= 1. For example, let *O*1(*η*, *ξ*) = *O*2(*η*, *ξ*) = *η* 2 *ξ* 2 , then (*O*<sup>1</sup> ~ *O*2)(*η*, *ξ*) = *η* 6 *ξ* 4 , we know that *O*<sup>1</sup> and *O*<sup>2</sup> satisfy (**HO-2**), i.e., *O*1(*αη*, *αξ*) = *α* <sup>2</sup>*O*1(*η*, *ξ*), but (*O*<sup>1</sup> ~*O*2)(*η*, *ξ*) does not satisfy (**HO-2**) since (*O*<sup>1</sup> ~*O*2)(*αη*, *αξ*) = *α* <sup>10</sup>*η* 6 *ξ* <sup>4</sup> <sup>6</sup><sup>=</sup> *<sup>α</sup>* 2*η* 6 *ξ* <sup>4</sup> = *α* 2 (*O*<sup>1</sup> ~ *O*2)(*η*, *ξ*).

The ~-composition does not preserve (**MI**). Assume that *O*<sup>1</sup> and *O*<sup>2</sup> satisfy (**MI**), then

$$\begin{array}{rcl} (O\_1 \circledast O\_2)(\eta, \alpha \xi) &= O\_1 \left( \eta, O\_2(\eta, \alpha \xi) \right) \\ &= O\_1 \left( \eta, O\_2(\alpha \eta, \xi) \right) \\ &\neq O\_1 \left( \alpha \eta, O\_2(\alpha \eta, \xi) \right) \\ &= (O\_1 \circledast O\_2)(\alpha \eta, \xi) \end{array}$$

for some *α*, *η*, *ξ* ∈ [0, 1].

The ~-composition does not preserve (*k***-LI**).

**Example 1.** *Let O*1(*η*, *ξ*) = *O*2(*η*, *ξ*) = *ηξ, then* (*O*<sup>1</sup> ~ *O*2)(*η*, *ξ*) = *η* 2 *ξ,*

$$\begin{array}{rcl} |O\_1(\eta\_1,\mathfrak{z}\_1) - O\_2(\eta\_2,\mathfrak{z}\_2)| &=& |\eta\_1\mathfrak{z}\_1 - \eta\_2\mathfrak{z}\_2| \\ &=& |\eta\_1\mathfrak{z}\_1 - \eta\_1\mathfrak{z}\_2 + \eta\_1\mathfrak{z}\_2 - \eta\_2\mathfrak{z}\_2| \\ &=& |\eta\_1(\mathfrak{z}\_1 - \mathfrak{z}\_2) + \mathfrak{z}\_2(\eta\_1 - \eta\_2)| \\ &\le& |\eta\_1(\mathfrak{z}\_1 - \mathfrak{z}\_2)| + |\mathfrak{z}\_2(\eta\_1 - \eta\_2)| \\ &\le& |\mathfrak{z}\_1 - \mathfrak{z}\_2| + |\eta\_1 - \eta\_2|. \end{array}$$

*Thus, O*<sup>1</sup> *and O*<sup>2</sup> *satisfy (***1-LI***). Let η*<sup>1</sup> = *ξ*<sup>1</sup> = 0.8 *and η*<sup>2</sup> = *ξ*<sup>2</sup> = 1, *then* (*O*<sup>1</sup> ~ *O*2)(0.8, 0.8) − (*O*<sup>1</sup> ~ *O*2)(1, 1) = 0.488 > 0.4 = (|0.8 − 1| + |0.8 − 1|)*, so O*<sup>1</sup> ~ *O*<sup>2</sup> *does not satisfy (***1-LI***).*

However, we have the following result.

**Theorem 7.** *If two overlap functions O*<sup>1</sup> *and O*<sup>2</sup> *respectively satisfy (k***1-LI***) and (k***2-LI***), then their* ~*-composition* (*O*<sup>1</sup> ~ *O*2) *satisfies (***(***k***<sup>1</sup>** *+ k***1***k***2)-LI***).*

**Proof.** Assume that *O*<sup>1</sup> and *O*<sup>2</sup> respectively satisfy (*k***1-LI**) and (*k***2-LI**), then, for any *η*1, *η*2, *ξ*1, *ξ*<sup>2</sup> ∈ [0, 1], we have

$$\begin{array}{c} \left| \left( O\_{1} \oplus O\_{2} \right) (\eta\_{1}, \mathfrak{z}\_{1}) - \left( O\_{1} \oplus O\_{2} \right) (\eta\_{2}, \mathfrak{z}\_{2}) \right| \\ \leq & \frac{1}{k\_{1}} \left( |\eta\_{1} - \eta\_{2}| + |O\_{2} \left( \eta\_{1}, \mathfrak{z}\_{1} \right) - O\_{2} \left( \eta\_{2}, \mathfrak{z}\_{2} \right) \right) \\ \leq & k\_{1} \left( |\eta\_{1} - \eta\_{2}| + |O\_{2} \left( \eta\_{1}, \mathfrak{z}\_{1} \right) - O\_{2} \left( \eta\_{2}, \mathfrak{z}\_{2} \right) \right) \\ \leq & k\_{1} \left( |\eta\_{1} - \eta\_{2}| + k\_{2} |\eta\_{1} - \eta\_{2}| + k\_{2} |\mathfrak{z}\_{1} - \mathfrak{z}\_{2}| \right) \\ = & (k\_{1} + k\_{1}k\_{2}) |\eta\_{1} - \eta\_{2}| + k\_{1}k\_{2} |\mathfrak{z}\_{1} - \mathfrak{z}\_{2}| \\ \leq & (k\_{1} + k\_{1}k\_{2}) \left( |\eta\_{1} - \eta\_{2}| + |\xi\_{1} - \xi\_{2}| \right). \end{array}$$

### *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.

**Table 2.** Properties preservation of the compositions.


### **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.


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.

**Author Contributions:** Funding acquisition, S.D. and Y.X.; Writing—original draft, S.D. and Y.X.; Writing—review and editing, L.D. and H.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Science Foundation of China (Grant Nos. 62006168 and 62101375) and Zhejiang Provincial Natural Science Foundation of China (Grant Nos. LQ21A010001 and LQ21F020001).

**Institutional Review Board Statement:** Not applicable

**Informed Consent Statement:** Not applicable

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

