Discrete Pseudo-Quasi Overlap Functions and Their Applications in Fuzzy Multi-Attribute Group Decision-Making

Abstract

The overlap function, a continuous aggregation function, is widely used in classification, decision-making, image processing, etc. Compared to applications, overlap functions have also achieved fruitful results in theory, such as studies on the fundamental properties of overlap functions, various generalizations of the concept of overlap functions, and the construction of additive and multiplicative generators based on overlap functions. However, most of the research studies on the overlap functions mentioned above contain commutativity and continuity, which can limit their practical applications. In this paper, we remove the symmetry and continuity from overlap functions and define discrete pseudo-quasi overlap functions on finite chains. Meanwhile, we also discuss their related properties. Then, we introduce pseudo-quasi overlap functions on sub-chains and construct discrete pseudo-quasi overlap functions on finite chains using pseudo-quasi overlap functions on these sub-chain functions. Unlike quasi-overlap functions on finite chains generated by the ordinal sum, discrete pseudo-quasi overlap functions on finite chains constructed through pseudo-quasi overlap functions on different sub-chains are dissimilar. Eventually, we remove the continuity from pseudo-automorphisms and propose the concept of pseudo-quasi-automorphisms. Based on this, we utilize pseudo-overlap functions, pseudo-quasi-automorphisms, and integral functions to obtain discrete pseudo-quasi overlap functions on finite chains, moreover, we apply them to fuzzy multi-attribute group decision-making. The results indicate that compared to overlap functions and pseudo-overlap functions, discrete pseudo-quasi overlap functions on finite chains have stronger flexibility and a wider range of practical applications.

1. Introduction

To establish a mathematical model of fuzzy objects, Zadeh proposed the concept of fuzzy sets [1] in 1965. Many scholars have conducted extensive research on the fuzzy set theory and applied it in pattern recognition, medical diagnosis, and fuzzy control [2,3,4,5]. In 1973, Zadeh proposed the famous CRI algorithm [6], which was a very effective tool for describing and dealing with the fuzziness of things and the uncertainty of systems, as well as for simulating human intelligence and decision-making. Fuzzy reasoning has been applied with great success in industrial control and manufacturing household appliances. However, compared with its application, the theoretical foundation of fuzzy reasoning is not flawless. In 1993, Elkan presented a report titled “The Seemingly Right Success of Fuzzy Logic” at the 11th Annual Conference on Artificial Intelligence [7], which caused a huge uproar. Many scholars have commented on this. Wu discussed this debate in [8]. Ying [9] pointed out that “although many of Erkan’s views are incorrect, and Wu has made some clarifications, we must also recognize that the lack of systematic and in-depth theoretical research in fuzzy logic is an undeniable fact.” Of course, there was no consensus on this debate. In fact, this debate has never been resolved. Meanwhile, it is precisely for this reason that fuzzy logic has become an active area of research, with many scholars achieving significant results in the field. In recent years, research on fuzzy sets has garnered widespread attention. Therefore, we delve into both the theoretical foundations and practical applications related to fuzzy sets.

In 2010, Bustine et al. proposed the definition of overlap functions [10]. As a special binary aggregation function, the overlap function has been widely used in decision-making, image processing, classification, and other fields [11,12,13]. Moreover, many academics have achieved significant outcomes in the theoretical research of overlap functions, specifically manifested in the following aspects: (1) research on basic properties of overlap functions, such as migrativity, homogeneity, Lipschitzianity, Archimedes, idempotence, etc. [14,15,16]; (2) extensions of various concepts related to overlap functions, including quasi-overlap functions [17], pseudo-overlap functions [18], semi-overlap functions [19], and so on [20,21,22]; (3) study of inducing various types of implication operators from overlap functions and group functions [23,24,25]; (4) construction of additive and multiplicative generators for overlap functions and various generalized overlap functions [26,27,28,29].

Aggregation is an important concept in decision theory, information fusion, and fuzzy inference systems. It involves converting several numerical values into a representative value; this process is called aggregation, and the function that executes this process is called an aggregation function. As powerful tools for processing information fusion, aggregation functions have been widely used in classification [30], fuzzy systems and control [31], hierarchical information fusion [32], and so on [33,34,35]. In order to better handle information fusion problems, many scholars have degenerated the aggregation functions (including t-norms, uninorms, t-operators, etc.) in [0, 1] to finite chains, and achieved relevant results [36,37,38]. Qiao transformed the overlap function and quasi-overlap function on [0, 1] into finite chains [39,40], and studied their related properties.

A fuzzy multi-attribute group decision-making problem can be described as a given set of possible alternative solutions, and each solution needs to be comprehensively evaluated from several attributes. Our goal is to find the optimal solution from this set of alternative solutions or to comprehensively rank this set of alternative solutions; the ranking results can reflect the decision-maker’s intention. The presence of uncertainty in fuzzy multi-attribute group decision-making processes can be represented by fuzzy sets. Therefore, fuzzy logic plays an important role in the field of fuzzy multi-attribute group decision-making. Fuzzy multi-attribute decision-making represents a non-classical approach to multi-attribute decision-making, extending and developing classical multi-attribute decision-making theories. Bass and Kwakernaak [41] proposed a method for addressing fuzzy multi-attribute group decision-making under uncertainty. Following their work, various scholars have proposed numerous types of fuzzy multi-attribute decision-making methods. Kichert, Zimmermann, and Chen et al. [42,43,44] summarized the above fuzzy multi-attribute decision-making methods. A few academics have also studied the application of overlap functions and certain generalized overlap functions in fuzzy multi-attribute group decision-making [45,46]. Mao et al. [47] proposed a fuzzy multi-attribute decision-making method based on the Sugeno integral semantics of overlap functions using fuzzy quantifiers, and verified the feasibility of this method through specific examples. Wen [48] combined overlap functions with rough sets to propose a new class of models, and then extended this model to multi-granularity, thereby establishing a solution method for fuzzy multi-attribute decision-making problems. Silva et al. [49] introduced a weighted average operator generated by n-dimensional overlap and aggregation functions, which they applied to fuzzy multi-attribute group decision-making problems. On this basis, Zhang et al. [18] extended the aforementioned weighted average operator and explored the use of pseudo-overlap functions in fuzzy multi-attribute group decision-making.

With the background information mentioned above and the current status of studying nationally as well as globally, the research motivations and innovation points of this paper are as follows:

(1) At present, most concepts of overlap functions and generalized overlap functions include symmetry and continuity, which can limit their practical applications. Thus, we remove the symmetry and continuity from overlap functions and introduce the concept of discrete pseudo-quasi overlap functions on finite chains. In addition, we have also studied their related properties.

(2) Currently, there is little research on constructing aggregate functions based on ordinal sums. Qiao [40] used ordinal sums to construct quasi-overlap functions on finite chains. This method constructs quasi-overlap functions on finite chains through quasi-overlap functions on sub-chains; each sub-chain is called an addend. Therefore, we naturally attempt to generalize the method of constructing quasi-overlap functions on the finite chains mentioned above and use a new method to construct discrete pseudo-quasi overlap functions on finite chains.

(3) In most literature (such as [18,47,49]), the aggregation functions used in the application of fuzzy multi-attribute group decision-making are all continuous. However, in the practical application of fuzzy multi-attribute group decision-making, the data objects involved are usually discrete. On the other hand, the discrete aggregation function has better flexibility and a wider range of applications in fuzzy multi-attribute group applications. Therefore, we apply discrete pseudo-quasi overlap functions on finite chains to fuzzy multi-attribute group decision-making. This approach not only promotes the development of fuzzy multi-attribute decision-making but also provides valuable reference and guidance for the theoretical development and practical application of overlap functions.

The main contents of this paper could be summarized as follows: In Section 2, we mainly present some basic knowledge on the topic. In Section 3, we introduce the concept of discrete pseudo-quasi overlap functions on finite chains and study their related properties. In Section 4, we offer pseudo-quasi overlap functions on sub-chains and construct discrete pseudo-quasi overlap functions on finite chains through pseudo-quasi overlap functions on sub-chains. Moreover, compared to quasi-overlap functions on finite chains generated by the ordinal sum, the discrete pseudo-quasi overlap functions on finite chains created by pseudo-quasi overlap functions on various sub-chains are different. In Section 5, we present the idea of pseudo-quasi-automorphism by removing the continuity from pseudo-automorphisms. Based on this, we use pseudo-overlap functions, pseudo-quasi-automorphisms, and integer functions to construct discrete pseudo-quasi overlap functions on finite chains and apply them to fuzzy multi-attribute group decision-making. The findings show that discrete pseudo-quasi overlap functions have better flexibility and adaptability than overlap functions and pseudo-overlap functions in applications. The research contents of this paper are shown in Figure 1.

Figure 1. Framework diagram of the paper.

2. Preliminaries

In this portion, we mainly provide some preliminary knowledge that is used in later sections.

Definition 1 

([10]). A binary function $O:{[0,1]}^2\to [0,1]$ is known as an overlap function if it meets $\forall x,y\in [0,1]$,

$\left(O1\right)$ 

O is symmetric;

$\left(O2\right)$ 

$O(x,y)=0\iff x=0$ or $y=0$;

$\left(O3\right)$ 

$O(x,y)=1\iff x=1$ and $y=1$;

$\left(O4\right)$ 

O is non-decreasing;

$\left(O5\right)$ 

O is continuous.

Definition 2 

([17]). A binary function $QO:{[0,1]}^2\to [0,1]$ is known as a quasi-overlap function if it satisfies $\left(O1\right)-\left(O4\right)$.

Definition 3 

([18]). A binary function $PO:{[0,1]}^2\to [0,1]$ is referred to as a pseudo-overlap function if it satisfies $\left(O2\right)-\left(O5\right)$.

Definition 4 

([20]). An n-ary function $O^n:{[0,1]}^n\to [0,1]$ is known as an n-ary overlap function if it meets $\forall x_{i+1},x_{i+2},x_{i+3},\dots ,x_{i+n}\in [0,1]$,

$\left(O^{n}1\right)$ 

$O^n$ is symmetry;

$\left(O^{n}2\right)$ 

$O^n(x_{i+1},x_{i+2},x_{i+3},\dots ,x_{i+n})=0\iff$ for $j\in N^{+},1\le j\le n$, such as ${\displaystyle \prod _{j=1}^n}{x}_{i+j}=0$;

$\left(O^{n}3\right)$ 

$O^n(x_{i+1},x_{i+2},x_{i+3},\dots ,x_{i+n})=1\iff$ for $j\in N^{+},1\le j\le n$, such as ${\displaystyle \prod _{j=1}^n}{x}_{i+j}=1$;

$\left(O^{n}4\right)$ 

$O^n$ is non-decreasing;

$\left(O^{n}5\right)$ 

$O^n$ is continuous.

Definition 5 

([18]). An n-ary function ${PO}^n:{[0,1]}^n\to [0,1]$ is known as an n-ary pseudo-overlap function if it satisfies $\left(O^{n}2\right)-\left(O^{n}5\right)$.

3. Discrete Pseudo-Quasi Overlap Functions

In this part, we delete the symmetry of quasi-overlap functions and introduce the notion of discrete pseudo-quasi overlap functions on finite chains. In addition, we discuss some of the associated properties, like Archimedean, idempotence, and cancellation law.

We define a finite chain $\mathcal{L}$ as follows:

Let $\mathcal{L}=\{x_0,x_1,x_2,\dots ,x_n,x_{n+1}\}$ be a set, $n\in N^{+},n\ge 1$. $\mathcal{L}$ is called a finite chain when it satisfies $\forall x_i,x_j\in \mathcal{L}$,

$\left(\mathcal{L}1\right)$ 

$x_i<x_j\iff i<j$;

$\left(\mathcal{L}2\right)$ 

$x_0$ is the minimum element, and $x_{n+1}$ is the maximum element of $\mathcal{L}$.

Definition 6. 

A binary function $PQ{O}_{\mathcal{L}}:{\mathcal{L}}^2\to \mathcal{L}$ is called a discrete pseudo-quasi overlap function on $\mathcal{L}$ if it fulfills $\forall x_i,x_j\in \mathcal{L}$,

$\left(PQ{O}_{\mathcal{L}}1\right)$ 

$PQ{O}_{\mathcal{L}}(x_i,x_j)=x_0\iff x_i=x_0$ or $x_j=x_0$;

$\left(PQ{O}_{\mathcal{L}}2\right)$ 

$PQ{O}_{\mathcal{L}}(x_i,x_j)=x_{n+1}\iff x_i=x_{n+1}$ and $x_j=x_{n+1}$;

$\left(PQ{O}_{\mathcal{L}}3\right)$ 

$PQ{O}_{\mathcal{L}}$ is non-decreasing.

A discrete pseudo-quasi overlap function $PQ{O}_{\mathcal{L}}$ is called an $x_{n+1}$-section left deflation on $\mathcal{L}$ when it satisfies $\forall x_i\in \mathcal{L}$,

• $\left(PQ{O}_{\mathcal{L}}4\right)$ $PQ{O}_{\mathcal{L}}(x_{n+1},x_i)\le x_i$.
• Correspondingly, a discrete pseudo-quasi overlap function $PQ{O}_{\mathcal{L}}$ is called an $x_{n+1}$-section right deflation on $\mathcal{L}$ when it satisfies $\forall x_i\in \mathcal{L}$,
• $\left(PQ{O}_{\mathcal{L}}5\right)PQ{O}_{\mathcal{L}}(x_i,x_{n+1})\le x_i$.

Assuming $\mathcal{L}=L=\{0,x_1,x_2,\dots x_n,1\}$ is a finite chain, we extend the L to [0, 1], then $PQ{O}_L$ is a pseudo-quasi overlap function given in [21]. On the other hand, a discrete pseudo-quasi overlap function $PQ{O}_L$ that satisfies symmetry is a quasi-overlap function $Q{O}_L$ on L, as mentioned in [40]. Moreover, the above $\left(PQ{O}_{\mathcal{L}}4\right)$ and $\left(PQ{O}_{\mathcal{L}}5\right)$ correspond to item (5) from Definition 2.1 in [40].

In the following sections, we use L to indicate the finite chain $\{0,x_1,x_2,\dots x_n,1\}$.

Below, we provide some examples of discrete pseudo-quasi overlap functions $PQ{O}_{\mathcal{L}}$ on $\mathcal{L}$.

Example 1.  (1) Let $\mathcal{L}$ be a finite chain. Then, for $n=1$, any discrete pseudo-quasi overlap function ${PQO}_{\mathcal{L}}$ is a quasi-overlap function ${QO}_{\mathcal{L}}$ on $\mathcal{L}$.

Taking $\mathcal{L}=\{{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{3}},and{\displaystyle \frac{1}{2}}\},n=1$. A graph of the $PQ{O}_{\mathcal{L}}$ is shown in Figure 2.

Figure 2. A discrete pseudo-quasi overlap function $PQ{O}_{\mathcal{L}}$.

(2) Let $L=\{0,x_1,x_2,\dots x_n,1\}$ be a finite chain, $n\in N^{+},n\ge 2$, $x_2,x_a,x_b\in L,x_2<x_a\le x_b$. Then, $\forall x_c,x_d\in L$, the function $PQ{O}_L:L^2\to L$, defined as follows:

 

$$PQ{O}_L(x_c,x_d)=\left\{\begin{array}{cc}{x}_1,\hfill & if{x}_1<x_c<x_a,x_1<x_d\le x_b\hfill \\ \min \{x_c,x_d\}.\hfill & otherwise\hfill \end{array}\right.$$

is a discrete pseudo-quasi overlap function on L.

Taking $L=\{0,0.1,0.2,,\dots ,0.9,1\},n=9,x_a=0.3,x_b=0.6$. A graph of the $PQ{O}_L$ is shown in Figure 3.

Figure 3. A discrete pseudo-quasi overlap function $PQ{O}_L$.

(3) Let $\mathcal{L}=L_N=\{0,1,2,3,\dots ,n+1\}$ be a finite chain with natural numbers, $n\in N^{+},n\ge 2$, $x_r,x_s\in L_N,1<x_r<x_s$. Then, $\forall x_p,x_q\in L_N$, the function $PQ{O}_{L_N}:L_N^2\to L_N$, defined as follows:

 

$${PQO}_{L_N}(x_p,x_q)=\left\{\begin{array}{cc}1,\hfill & if0<x_p<x_r,0<x_q<x_s\hfill \\ \left[\sqrt{x_p{x}_q}\right].\hfill & otherwise\hfill \end{array}\right.$$

is a discrete pseudo-quasi overlap function on $L_N$.

Taking $L_N=\{0,1,2,,\dots ,9,10\},n=9,x_r=2,x_s=6$. An image of the $PQ{O}_{L_N}$ is shown in Figure 4.

Figure 4. A discrete pseudo-quasi overlap function ${PQO}_{L_N}$.

$\left(4\right)$ Let $\mathcal{L}=L_{N^{+}}=\{1,2,3,\dots ,n+1\}$ be a finite chain with positive integers, $n\in N^{+},n\ge 2$, $x_g,x_h\in L_{N^{+}},x_g\ne x_h,2<\min \{x_g,x_h\}$. Then, $\forall x_m,x_n\in L_{N^{+}}$, the function ${PQO}_{L_{N^{+}}}:L_{N^{+}}^2\to L_{N^{+}}$, defined as follows:

 

$${PQO}_{L_{N^{+}}}(x_m,x_n)=\left\{\begin{array}{cc}1,\hfill & if{x}_m=1or{x}_n=1\hfill \\ 2,\hfill & if1<x_m\le x_g,1<x_n\le x_h\hfill \\ \left[{\displaystyle \frac{2x_m{x}_n}{x_m+x_n}}\right].\hfill & otherwise\hfill \end{array}\right.$$

is a discrete pseudo-quasi overlap function on $L_{N^{+}}$.

Taking $L_N^{+}=\{1,2,,\dots ,9,10\},n=8,x_g=4,x_h=6$. An image of the ${PQO}_{L_N}$ is shown in Figure 5.

Figure 5. A discrete pseudo-quasi overlap function ${PQO}_{L_{N^{+}}}$.

We observe that “$\left[x\right]$” in Example 1 is an integral function; more precisely, it is a round function to the nearest integer x. Furthermore, regarding other types of integral functions, such as floor, ceil, and fix, their methods of constructing discrete pseudo-quasi overlap functions on $\mathcal{L}$ are similar to that of the round function.

Next, we investigate the relevant properties of discrete pseudo-quasi overlap functions on finite chains L.

3.1. Archimedes of Discrete-Pseudo-Quasi Overlap Functions

First, we discuss the Archimedes of discrete pseudo-quasi overlap functions on L.

Definition 7. 

Let ${PQO}_L:L^2\to L$ be a discrete pseudo-quasi overlap function on L. ${PQO}_L$ is called Archimedean when it satisfies $\forall x_i,x_j\in L-\{0,1\},{\left(x_i\right)}_{{PQO}_L}^{\left(n\right)}<x_j$, and ${PQO}_L$ is given by the following:

 

$${\left(x_i\right)}_{{PQO}_L}^{\left(1\right)}=x_i,{\left(x_i\right)}_{{PQO}_L}^{(n+1)}={PQO}_L(x_i,{\left(x_i\right)}_{{PQO}_L}^n),n\in N^{+}.$$

Proposition 1. 

Let ${PQO}_L:L^2\to L$ be a discrete pseudo-quasi overlap function on L. Then, $\forall x_i\in L-\left\{1\right\},x_j\in L,{PQO}_L(x_i,x_j)$ is not strictly increasing.

Proof. 

We take $x_i=0,x_j,x_z\in L$, and $x_j<x_z$. Then, ${PQO}_L(x_i,x_j)=0\le 0={PQO}_L(x_i,x_z)$. So, $x_i=0$, $x_j\in L$, and $PQ{O}_L(x_i,x_j)$ is not strictly increasing. On the other hand, $x_i=1$, $x_j\in L$, we need to verify that $PQ{O}_L(x_i,x_j)$ is strictly increasing. For $x_j,x_z\in L$, and $x_j<x_z$, there are three different cases, as follows:

(1)

$x_i=1,x_j=0<x_z,PQ{O}_L(x_i,x_j)=0<PQ{O}_L(x_i,x_z)$;

(2)

$x_i=1,x_j<x_z=1,PQ{O}_L(x_i,x_j)<1=PQ{O}_L(x_i,x_z)$;

(3)

$x_i=1,x_j\in L-\left\{0\right\},x_z\in L-\left\{1\right\}$. Suppose that ${PQO}_L(x_i,x_j)$ is not strictly increasing. According to $\left({PQO}_{L}3\right)$, we know that ${PQO}_L(x_i,x_j)={PQO}_L(x_i,x_z)$. Obviously, this is contradictory to $x_j<x_z$. Thus, $x_i=1,x_j\in L$, ${PQO}_L(x_i,x_j)$ is strictly increasing. Finally, for the scenario where $x_i\in L-\{0,1\},x_j\in L$, and ${PQO}_L(x_i,x_j)$ is not strictly increasing, the proof method is similar to [34]. To summarize, $\forall x_i\in L-\left\{1\right\},x_j\in L,{PQO}_L(x_i,x_j)$ is not strictly increasing. □

From Proposition 1, we can immediately deduce that $\forall x_j\in L-\left\{1\right\},x_i\in L,PQ{O}_L(x_j,x_i)$ is also not strictly increasing.

Proposition 2. 

Let ${PQO}_L:L^2\to L$ be Archimedean. If ${PQO}_L$ is a discrete pseudo-quasi overlap function on L, then ${\left(x_i\right)}_{{PQO}_L}^{(n+1)}\le {\left(x_i\right)}_{{PQO}_L}^{\left(n\right)},n\in N^{+}$.

Proof. 

The following can directly be obtained through Definition 7, Proposition 1, and mathematical methods of induction: $\left(1\right)$ For $n=1,{\left(x_i\right)}_{{PQO}_L}^{\left(2\right)}={PQO}_L(x_i,x_i)<x_i={\left(x_i\right)}_{{PQO}_L}^{\left(1\right)}$, that is, ${\left(x_i\right)}_{{PQO}_L}^{\left(2\right)}\le {\left(x_i\right)}_{{PQO}_L}^{\left(1\right)}$. For $n=2,{\left(x_i\right)}_{{PQO}_L}^{\left(3\right)}={PQO}_L(x_i,{\left(x_i\right)}_{{PQO}_L}^{\left(2\right)})={PQO}_L(x_i,{PQO}_L(x_i,x_i))$. According to ${PQO}_L(x_i,x_i)<x_i$ and Proposition 1, we have the following:

$${PQO}_L(x_i,{PQO}_L(x_i,x_i))\le {PQO}_L(x_i,x_i).$$

Thus, ${\left(x_i\right)}_{{PQO}_L}^{\left(3\right)}\le {\left(x_i\right)}_{{PQO}_L}^{\left(2\right)}$. Assume that $n=k$, ${\left(x_i\right)}_{{PQO}_L}^{(k+1)}\le {\left(x_i\right)}_{{PQO}_L}^{\left(k\right)}$. For $n=k+1$, we have the following:

$${\left(x_i\right)}_{{PQO}_L}^{(k+2)}={PQO}_L(x_i,{\left(x_i\right)}_{{PQO}_L}^{(k+1)})\le {PQO}_L(x_i,{\left(x_i\right)}_{{PQO}_L}^{\left(k\right)})={\left(x_i\right)}_{{PQO}_L}^{(k+1)}.$$

Therefore, $n\in N^{+}$, ${\left(x_i\right)}_{{PQO}_L}^{(n+1)}\le {\left(x_i\right)}_{{PQO}_L}^{\left(n\right)}$. □

Proposition 3. 

Let ${PQO}_L:L^2\to L$ be a discrete pseudo-quasi overlap function on L. Then, ${PQO}_L$ is not Archimedean.

Proof. 

Suppose that ${PQO}_L$ is Archimedean. Owing to Proposition 2, we know the following:

$$0<\cdots \le {\left(x_i\right)}_{{PQO}_L}^{(n+1)}\le {\left(x_i\right)}_{{PQO}_L}^{\left(n\right)}\le {\left(x_i\right)}_{{PQO}_L}^{(n-1)}<\cdots \le {\left(x_i\right)}_{{PQO}_L}^{\left(2\right)}\le {\left(x_i\right)}_{{PQO}_L}^{\left(1\right)}=x_i,n\in N^{+}.$$

Thus, for $n\in N^{+},\underset{n\to \infty }{lim}{\left(x_i\right)}_{{PQO}_L}^{\left(n\right)}=0$, conflicting with ${\left(x_i\right)}_{{PQO}_L}^{\left(n\right)}<x_j$ for Definition 7. Therefore, ${PQO}_L$ is not Archimedean. □

Now, we will discuss the idempotence of discrete pseudo-quasi overlap functions on L.

3.2. Idempotence of Discrete Pseudo-Quasi Overlap Functions

Definition 8. 

Let ${PQO}_L:L^2\to L$ be a discrete pseudo-quasi overlap function on L. An element $x_i\in L$ is called idempotent when it satisfies ${PQO}_L(x_i,x_i)=x_i$. A discrete pseudo-quasi overlap function ${PQO}_L$ is called idempotent when it satisfies that $\forall x_i\in L$ is an idempotent element.

Obviously, 0 and 1 are idempotent elements of a discrete pseudo-quasi overlap function on L. Moreover, only the discrete pseudo-quasi overlap function $PQ{O}_{\mathcal{L}}$ on $\mathcal{L}$ generated by Case (1) in Example 1 is idempotent.

Proposition 4. 

Let ${PQO}_L:L^2\to L$ be a discrete pseudo-quasi overlap function on L. Then, there exists $x_i\in L-\{0,1\}$, such that ${PQO}_L(x_i,x_i)=x_i$.

Proof. 

The proof is analogous to [16]. □

Proposition 5. 

Let ${PQO}_L:L^2\to L$ be a discrete pseudo-quasi overlap function on L.

(1)

If ${PQO}_L$ satisfies $\left({PQO}_{L}4\right)$, then there exists $x_i\in L-\left\{0,1\right\}$, such that ${PQO}_L(1,x_i)<x_i$.

(2)

If ${PQO}_L$ satisfies $\left({PQO}_{L}5\right)$, then there exists $x_i\in L-\left\{0,1\right\}$, such that ${PQO}_L(x_i,1)<x_i$.

Proof. 

(1) Suppose that ${PQO}_L$ is a discrete pseudo-quasi overlap function on L. If ${PQO}_L$ satisfies $\left({PQO}_{L}4\right),$ then $\forall x_i\in L-\{0,1\}$, ${PQO}_L(1,x_i)\le x_i$. Moreover, according to Proposition 4 and $\left({PQO}_{L}3\right)$, we known that there exists $x_i\in L-\{0,1\}$, such that we have the following:

$$x_i={PQO}_L(x_i,x_i)\le {PQO}_L(1,x_i).$$

So, ${PQO}_L(1,x_i)<x_i$. The proofs of (2) are similar to (1). □

Proposition 6. 

Let ${PQO}_L:L^2\to L$ be a discrete pseudo-quasi overlap function on L. If ${PQO}_L$ is Archimedean, then ${PQO}_L$ has no idempotent element, except for $0,1$.

Proof. 

Suppose that there exists $x_i\in L-\{0,1\}$, which is an idempotent element of the discrete pseudo-quasi overlap function $PQ{O}_L$ on L. As ${PQO}_L$ is Archimedean, then $n=1,{\left(x_i\right)}_{{PQO}_L}^{\left(1\right)}=x_i$, and $n=2,{\left(x_i\right)}_{{PQO}_L}^{\left(2\right)}={PQO}_L(x_i,x_i)=x_i$. Assume that $n=k;$ we have the following:

$${\left(x_i\right)}_{{PQO}_L}^{\left(k\right)}={PQO}_L(x_i,{\left(x_i\right)}_{{PQO}_L}^{(k-1)})={PQO}_L(x_i,x_i)=x_i.$$

So, for $n=k+1,{\left(x_i\right)}_{{PQO}_L}^{(k+1)}={PQO}_L(x_i,{\left(x_i\right)}_{{PQO}_L}^{\left(k\right)})={PQO}_L(x_i,x_i)=x_i$. Thus, for $n\in N^{+},{\left(x_i\right)}_{{PQO}_L}^{\left(n\right)}=x_i$, which conflicts with ${\left(x_i\right)}_{{PQO}_L}^{\left(n\right)}<x_j$ in Definition 7. Thus, ${PQO}_L$ has no idempotent element, except for $0,1$. □

In the end, we discuss the cancellation law of discrete pseudo-quasi overlap functions on L.

3.3. Cancellation Law of Discrete Pseudo-Quasi Overlap Functions

Definition 9. 

Let ${PQO}_L:L^2\to L$ be a discrete pseudo-quasi overlap function on L. $\forall x_i,x_j,x_z\in L$, ${PQO}_L$ is said to fulfill the left-cancellation law if we have the following:

$${PQO}_L(x_i,x_j)={PQO}_L(x_i,x_z)\mathit{means}\mathit{that}{x}_i=x_0\mathit{or}{x}_j=x_z.$$

Similarly, ${PQO}_L$ is said to fulfill the right-cancellation law if we have the following:

$${PQO}_L(x_j,x_i)={PQO}_L(x_z,x_i)\mathit{means}\mathit{that}{x}_i=x_0\mathit{or}{x}_j=x_z.$$

Lemma 1. 

A discrete pseudo-quasi overlap function ${PQO}_L$ on L fulfills the cancellation law if $\forall x_i,x_j,x_z\in L$ satisfy the following conditions:

(1)

${PQO}_L(x_i,x_j)={PQO}_L(x_i,x_z)\Rightarrow x_i=x_0$ or $x_j=x_z$;

(2)

${PQO}_L(x_j,x_i)={PQO}_L(x_z,x_i)\Rightarrow x_i=x_0$ or $x_j=x_z$.

Proposition 7. 

Let ${PQO}_L:L^2\to L$ be a discrete pseudo-quasi overlap function on L.

(1)

${PQO}_L$ fulfills the left-cancellation law ⇔ for $\forall x_i\in L-\left\{0\right\},x_j\in L$, and ${PQO}_L(x_i,x_j)$ is strictly increasing.

(2)

${PQO}_L$ fulfills the right-cancellation law ⇔ for $\forall x_j\in L-\left\{0\right\},x_i\in L$, and ${PQO}_L(x_j,x_i)$ is strictly increasing.

Proof. 

The proofs of (1) and (2) are similar. Next, we only prove (2). (2) (Necessity) Suppose that ${PQO}_L$ fulfills the right-cancellation law, $i\ne 0,x_j<x_z$. Since ${PQO}_L$ is monotonically increasing, ${PQO}_L(x_j,x_i)\le {PQO}_L(x_z,x_i)$. We consider ${PQO}_L(x_j,x_i)={PQO}_L(x_z,x_i)$. Since ${PQO}_L$ fulfills the right-cancellation law, $x_i=0$ or $x_j=x_z$. Obviously, this contradicts with $x_j<x_z$. Thus, $PQ{O}_L(x_j,x_i)<PQ{O}_L(x_z,x_i)$. (Sufficiency) Suppose that $\forall x_j\in L-\left\{0\right\},x_i\in L,PQ{O}_L(x_j,x_i)$ is strictly increasing. So, $x_j<x_z$, and $PQ{O}_L(x_j,x_i)<PQ{O}_L(x_z,x_i)$. We assume that $PQ{O}_L$ does not fulfill the right-cancellation law. Then, if $PQ{O}_L(x_j,x_i)={PQO}_L(x_z,x_i)$, it means that $x_i\ne 0$ and $x_j\ne x_z$. So, $x_j<x_z$ or $x_j>x_z$ or $x_j\left|\right|x_z$. We consider $x_j<x_z$. Because ${PQO}_L(x_j,x_i)$ is strictly increasing, $PQ{O}_L(x_j,x_i)<PQ{O}_L(x_z,x_i)$. This contradicts with $PQ{O}_L(x_j,x_i)=PQ{O}_L(x_z,x_i)$. Thus, the scenario where $x_j<x_z$ does not exist. Similarly, the scenario where $x_j>x_z$ is also not valid. Finally, we consider $x_j\left|\right|x_z$. Apparently, this contradicts the premise that $PQ{O}_L(x_j,x_i)$ is strictly increasing. Therefore, $PQ{O}_L$ fulfills the right-cancellation law. □

Proposition 8. 

Let ${PQO}_L:L^2\to L$ be a discrete pseudo-quasi overlap function on L. Then, ${PQO}_L$ does not fulfill the cancellation law.

Proof. 

The proof is analogous to [17]. □

According to Propositions 4 and 8, we know that both discrete pseudo-quasi overlap functions and quasi-overlap functions on L can obtain similar conclusions. That is to say, symmetry does not significantly affect the conclusions of Propositions 4 and 8.

4. The Construction of Discrete Pseudo-Quasi Overlap Functions

In [40], Qiao proposed a method for constructing quasi-overlap functions on finite chains based on the ordinal sum. This method mainly utilizes quasi-overlap functions on sub-chains to create quasi-functions on finite chains, where these sub-chains are additive. Therefore, we extend this approach and devise a new method to construct discrete pseudo-quasi overlap functions. First, we define discrete pseudo-quasi overlap functions on a sub-chain. We then construct a discrete pseudo-quasi overlap function on finite chains by leveraging pseudo-quasi overlap functions on these sub-chains.

Definition 10. 

Let L be a finite chain, and $L_k^{*}=\left\{[x_k,x_{k+3}]\right|k\in N,0\le k\le n-2\}$ be a sub-chain of L. A binary function ${PQO}_{L_k^{*}}:{[x_k,x_{k+3}]}^2\to [x_k,x_{k+3}]$ is called a discrete pseudo-quasi overlap function on $L_k^{*}$ when it satisfies $\forall x_{\mu },x_{\nu }\in [x_k,x_{k+3}],$

$\left({PQO}_{L_k^{*}}1\right)$

${PQO}_{L_k^{*}}(x_{\mu },x_{\nu })=x_k\iff x_{\mu }=x_k$ or $x_{\nu }=x_k$;

$\left({PQO}_{L_k^{*}}2\right)$

${PQO}_{L_k^{*}}(x_{\mu },x_{\nu })=x_{k+3}\iff x_{\mu }=x_{k+3}$ and $x_{\nu }=x_{k+3}$;

$\left({PQO}_{L_k^{*}}3\right)$

${PQO}_{L_k^{*}}$ is non-decreasing.

Theorem 1. 

Let L be a finite chain, $k\in N,0\le k\le n-2,[x_k,x_{k+3}]$ be a sub-chain of L, and ${PQO}_{L_k^{*}}:{[x_k,x_{k+3}]}^2\to [x_k,x_{k+3}]$ be a discrete pseudo-quasi overlap function on $L_k^{*}$. Then, $\forall x_i,x_j\in L$, the function ${PQO}_{L_k}:L^2\to L$, is defined as follows:

$${PQO}_{L_k}(x_i,x_j)=\left\{\begin{array}{cc}{PQO}_{L_k^{*}}(x_i,x_j),\hfill & if{x}_i,x_j\in [x_k,x_{k+3}]\hfill \\ \min \{{\alpha }_1\left(x_i\right),{\alpha }_2\left(x_j\right)\}.\hfill & otherwise\hfill \end{array}\right.$$

is a discrete pseudo-quasi overlap function on L, among these, with different values of k, the functions ${\alpha }_1\left(x_i\right):L\to L$ and ${\alpha }_2\left(x_j\right):L\to L$ have different forms, as follows:

$$\left(i\right)k=0,n\ge 4.{\alpha }_1^{\left(1\right)}\left(x_i\right):L\to L,{\alpha }_2^{\left(1\right)}\left(x_j\right):L\to L,\mathit{separately},\mathit{given}\mathit{by}\mathit{the}\mathit{following}:$$

$${\alpha }_1^{\left(1\right)}\left(x_i\right)=\left\{\begin{array}{cc}{PQO}_{L_0^{*}}(x_i,x_3),\hfill & if{x}_i\in [x_0,x_3]\hfill \\ x_i.\hfill & otherwise\hfill \end{array}\right.$$

$${\alpha }_2^{\left(1\right)}\left(x_j\right)=\left\{\begin{array}{cc}{x}_w,\hfill & if{x}_j\in (x_0,x_3]\hfill \\ x_j.\hfill & otherwise\hfill \end{array}\right.$$

where $w\in N^{+},3\le w\le n$.

$$\left(ii\right)1\le k<n-2,n\ge 5;{\alpha }_1^{\left(2\right)}\left(x_i\right):L\to L,{\alpha }_2^{\left(2\right)}\left(x_j\right):L\to L,\mathit{are}\mathit{defined}\mathit{separately},\mathit{as}\mathit{follows}:$$

$${\alpha }_1^{\left(2\right)}\left(x_i\right)=\left\{\begin{array}{cc}{PQO}_{L_k^{*}}(x_i,x_{k+3}),\hfill & if{x}_i\in [x_k,x_{k+3}]\hfill \\ x_i.\hfill & otherwise\hfill \end{array}\right.$$

$${\alpha }_2^{\left(2\right)}\left(x_j\right)=\left\{\begin{array}{cc}{x}_t,\hfill & if{x}_j\in [x_k,x_{k+3}]\hfill \\ x_j.\hfill & otherwise\hfill \end{array}\right.$$

where $t\in N^{+},k+3\le t\le n$.

$$\left(iii\right)k=n-2,n\ge 4;{\alpha }_1^{\left(3\right)}\left(x_i\right):L\to L,{\alpha }_2^{\left(3\right)}\left(x_j\right):L\to L,\mathit{are}\mathit{defined}\mathit{separately},\mathit{as}\mathit{follows}:$$

$${\alpha }_1^{\left(3\right)}\left(x_i\right)=\left\{\begin{array}{cc}{PQO}_{L_k^{*}}(x_i,x_{n+1}),\hfill & if{x}_i\in [x_{n-2},x_{n+1}]\hfill \\ x_{{\gamma }_{\theta }}\hfill & if{x}_i=x_{{\beta }_{\theta }}\hfill \\ x_i.\hfill & otherwise\hfill \end{array}\right.$$

$${\alpha }_2^{\left(3\right)}\left(x_j\right)=\left\{\begin{array}{cc}{x}_{\lambda },\hfill & if{x}_j\in [x_{n-2},x_{n+1}]\hfill \\ x_j\hfill & otherwise\hfill \end{array}\right.$$

for $\theta \in N^{+},1\le \theta ,{\beta }_{\theta }\le k-1$, $1\le {\gamma }_{\theta },\lambda \le k$, such as ${\gamma }_{\theta }\le \lambda$. ${\beta }_{\theta }$ and ${\gamma }_{\theta }$ are in one-to-one correspondence. The details are as follows:

$$\begin{array}{cc}{\gamma }_1=k(ork-2);\hfill & if{\beta }_1=k-1,\hfill \\ {\gamma }_2=k-1(ork-3);\hfill & if{\beta }_2=k-2,\hfill \\ \dots \hfill & \dots \hfill \\ {\gamma }_{k-2}=3\left(or1\right);\hfill & if{\beta }_{k-2}=2,\hfill \\ {\gamma }_{k-1}=2;\hfill & if{\beta }_{k-1}=1,\hfill \end{array}$$

Proof. 

$\left(i\right),\left(ii\right),$ and $\left(iii\right)$ are similar. Next, we only prove $\left(iii\right)$. Without loss of generality, we take $\theta =1,{\beta }_1=n-3,{\gamma }_1=\lambda =n-2$. $\left(PQ{O}_{L}1\right)$ (Necessity) If ${PQO}_{L_{n-2}}(x_i,x_j)=x_0$, then $\min \{{\alpha }_1^{\left(3\right)}\left(x_i\right),{\alpha }_2^{\left(3\right)}\left(x_j\right)\}=x_0$, i.e., ${\alpha }_1^{\left(3\right)}\left(x_i\right)=x_0$ or ${\alpha }_2^{\left(3\right)}\left(x_j\right)=x_0$. According to $k\in N,k=n-2$, we know that $x_i\notin [x_{n-2},x_{n+1}]\cup \left\{x_{n-3}\right\}$. So, ${\alpha }_1^{\left(3\right)}\left(x_i\right)=x_i=x_0$. On the other hand, $x_j\notin [x_{n-2},x_{n+1}]$. So, ${\alpha }_2^{\left(3\right)}\left(x_j\right)=x_j=x_0$. Thus, $x_i=x_0$ or $x_j=x_0$. (Sufficiency) If $x_i=x_0$ or $x_j=x_0$. Without loss of generality, we take $x_i=x_0$. Since $x_i\notin [x_{n-2},x_{n+1}]\cup \left\{x_{n-3}\right\}$, we obtain ${\alpha }_1^{\left(3\right)}\left(x_i\right)={\alpha }_1^{\left(3\right)}\left(x_0\right)=x_0$. So, we have the following:

$${PQO}_{L_{n-2}}(x_i,x_j)=\min \{{\alpha }_1^{\left(3\right)}\left(x_i\right),{\alpha }_2^{\left(3\right)}\left(x_j\right)\}=\min \{x_0,{\alpha }_2^{\left(3\right)}\left(x_j\right)\}=x_0.$$

On the other hand, if $x_j=x_0$, and $x_j\notin [x_{n-2},x_{n+1}]$, we also obtain ${\alpha }_2^{\left(3\right)}\left(x_j\right)={\alpha }_2^{\left(3\right)}\left(x_0\right)=x_0$. So, ${PQO}_{L_{n-2}}(x_i,x_j)=\min \{{\alpha }_1^{\left(3\right)}\left(x_i\right),{\alpha }_2^{\left(3\right)}\left(x_j\right)\}=\min \{{\alpha }_1^{\left(3\right)}\left(x_i\right),x_0\}=x_0$. Thus, we have the following:

$${PQO}_{L_{n-2}}(x_i,x_j)=x_0.$$

Therefore, ${PQO}_{L_{n-2}}$ satisfies $\left(PQ{O}_{L}1\right)$.

$\left(PQ{O}_{L}2\right)$ (Necessity) If ${PQO}_{L_{n-2}}(x_i,x_j)=x_1$, then $\min \{{\alpha }_1^{\left(3\right)}\left(x_i\right),{\alpha }_2^{\left(3\right)}\left(x_j\right)\}=x_1$, that is, ${\alpha }_1^{\left(3\right)}\left(x_i\right)=x_1$ and ${\alpha }_2^{\left(3\right)}\left(x_j\right)=x_1$. For $k\in N,k=n-2$, we know that $x_i\notin [x_{n-2},x_{n+1}]\cup \left\{x_{n-3}\right\}$, and $x_j\notin [x_{n-2},x_{n+1}]$. So, $x_i={\alpha }_1^{\left(3\right)}\left(x_i\right)=x_1$, and $x_j={\alpha }_2^{\left(3\right)}\left(x_j\right)=x_1$. (Sufficiency) If $x_i=x_1$ and $x_j=x_1$. Since $k=n-2$, $k\in N$, we gain $x_i\notin [x_{n-2},x_{n+1}]\cup \left\{x_{n-3}\right\}$, and $x_j\notin [x_{n-2},x_{n+1}]$. So, ${\alpha }_1^{\left(3\right)}\left(x_i\right)=x_i=x_1$, and ${\alpha }_2^{\left(3\right)}\left(x_j\right)=x_j=x_1$. Thus, we have the following:

$${PQO}_{L_{n-2}}(x_i,x_j)=\min \{{\alpha }_1^{\left(3\right)}\left(x_i\right),{\alpha }_2^{\left(3\right)}\left(x_j\right)\}=\min \{x_1,x_1\}=x_1.$$

Therefore, ${PQO}_{L_{n-2}}$ satisfies $\left(PQ{O}_{L}2\right)$.

$\left(PQ{O}_{L}3\right)$ $\forall x_i,x_j,x_z\in L,x_j\le x_z$, we have several situations, specifically as follows:

(1) $x_i,x_j,x_z\in [x_{n-2},x_{n+1}]$. Since that ${PQO}_{L_{n-2}^{*}}$ is a discrete pseudo-quasi overlap function on $L_{n-2}^{*}$. So, ${PQO}_{L_{n-2}}(x_i,x_j)={PQO}_{L_{n-2}^{*}}(x_i,x_j)\le {PQO}_{L_{n-2}^{*}}(x_i,x_z)={PQO}_{L_{n-2}}(x_i,x_z)$.

(2) $x_i,x_j,x_z\notin [x_{n-2},x_{n+1}]$ at the same time.

(2.1) $x_i,x_z\in [x_{n-2},x_{n+1}],x_j\notin [x_{n-2},x_{n+1}]$, without loss of generality, we take $x_j=x_{n-3}$. Then, ${PQO}_{L_{n-2}}(x_i,x_j)=\min \{{\alpha }_1^{\left(3\right)}\left(x_i\right),{\alpha }_2^{\left(3\right)}\left(x_j\right)\}=\min \{{PQO}_{L_{n-2}^{*}}(x_i,x_{n+1}),x_{n-3}\}$, and ${PQO}_{L_{n-2}}(x_i,x_z)={PQO}_{L_{n-2}^{*}}(x_i,x_z)$. Since $x_{n-3}<x_{n-2}\le {PQO}_{L_k^{*}}(x_i,x_{n+1})\le x_{n+1}$, we have the following:

$${PQO}_{L_n}(x_i,x_j)=\min \{{PQO}_{L_n^{*}}(x_i,x_{n+1}),x_{n-3}\}=x_{n-3}.$$

According to $x_{n-3}<x_{n-2}\le PQ{O}_{L_{n-2}^{*}}(x_i,x_z)$, we know that $PQ{O}_{L_{n-2}}(x_i,x_j)\le PQ{O}_{L_{n-2}}(x_i,x_z)$.

(2.2) $x_i\in [x_{n-2},x_{n+1}],x_j,x_z\notin [x_{n-2},x_{n+1}]$, without loss of generality, we take $x_j=x_{n-4},x_z=x_{n-3}$. Then, $PQ{O}_{L_{n-2}}(x_i,x_j)=\min \{PQ{O}_{L_{n-2}^{*}}(x_i,x_{n+1}),x_{n-4}\}$, and we have the following:

$${PQO}_{L_{n-2}}(x_i,x_z)=\min \{{PQO}_{L_{n-2}^{*}}(x_i,x_{n+1}),x_{n-3}\}.$$

Thus, ${PQO}_{L_{n-2}}(x_i,x_j)\le {PQO}_{L_{n-2}}(x_i,x_z)$.

(2.3) $x_i=x_{n-3},x_j,x_z\in [x_{n-2},x_{n+1}]$. Then, we have the following:

$${PQO}_{L_{n-2}}(x_i,x_j)=\min \{x_{n-2},x_{n-2}\}=x_{n-2},$$

and ${PQO}_{L_{n-2}}(x_i,x_z)=\min \{x_{n-2},x_{n-2}\}=x_{n-2}$. So, ${PQO}_{L_{n-2}}(x_i,x_j)\le {PQO}_{L_{n-2}}(x_i,x_z)$.

(2.4) $x_i=x_{n-3},x_j\notin [x_{n-2},x_{n+1}]$, without loss of generality, we take $x_j=x_{n-3}$, $x_z\in [x_{n-2},x_{n+1}]$. Then, ${PQO}_{L_{n-2}}(x_i,x_j)=\min \{x_{n-2},x_{n-3}\}=x_{n-3}$, and we have the following:

$${PQO}_{L_{n-2}}(x_i,x_z)=\min \{x_{n-2},x_{n-2}\}=x_{n-2}.$$

So, ${PQO}_{L_{n-2}}(x_i,x_j)\le {PQO}_{L_{n-2}}(x_i,x_z)$.

(2.5) $x_i=x_{n-3},x_j,x_z\notin [x_{n-2},x_{n+1}]$, without loss of generality, we take $x_j=x_z=x_{n-3}$. Then, ${PQO}_{L_{n-2}}(x_i,x_j)=\min \{x_{n-2},x_{n-3}\}=x_{n-3}$, and we have the following:

$${PQO}_{L_{n-2}}(x_i,x_z)={PQO}_{L_{n-2}}(x_i,x_j)=x_{n-3}.$$

So, ${PQO}_{L_{n-2}}(x_i,x_j)\le {PQO}_{L_{n-2}}(x_i,x_z)$.

(2.6) $x_i\notin [x_{n-2},x_{n+1}]\cup \left\{x_{n-3}\right\}$, without loss of generality, we take $x_i=x_{n-4}$, $x_j,x_z\in [x_{n-2},x_{n+1}]$. Then, ${PQO}_{L_{n-2}}(x_i,x_j)=\min \{x_{n-4},x_{n-2}\}=x_{n-4}$, and we have the following:

$${PQO}_{L_{n-2}}(x_i,x_z)={PQO}_{L_{n-2}}(x_i,x_j)=x_{n-4}.$$

So, ${PQO}_{L_{n-2}}(x_i,x_j)\le {PQO}_{L_{n-2}}(x_i,x_z)$.

(2.7) $x_i\notin [x_{n-2},x_{n+1}]\cup \left\{x_{n-3}\right\},x_j\notin [x_{n-2},x_{n+1}]$, without loss of generality, we take $x_i=x_{n-4},x_j=x_{n-5},x_z\in [x_{n-2},x_{n+1}]$. Then, ${PQO}_{L_{n-2}}(x_i,x_j)=\min \{x_{n-4},x_{n-5}\}=x_{n-5}$, and ${PQO}_{L_{n-2}}(x_i,x_z)=\min \{x_{n-4},x_{n-2}\}=x_{n-4}$. So, ${PQO}_{L_{n-2}}(x_i,x_j)\le {PQO}_{L_{n-2}}(x_i,x_z)$.

(2.8) $x_i\notin [x_{n-2},x_{n+1}]\cup \left\{x_{n-3}\right\},x_j,x_z\notin [x_{n-2},x_{n+1}]$, without loss of generality, we take $x_i=x_{n-4},x_j=x_z=x_{n-5}$. Then, ${PQO}_{L_{n-2}}(x_i,x_j)=\min \{x_{n-4},x_{n-5}\}=x_{n-5}$, and ${PQO}_{L_{n-2}}(x_i,x_z)={PQO}_{L_{n-2}}(x_i,x_j)=x_{n-5}$. So, ${PQO}_{L_{n-2}}(x_i,x_j)\le {PQO}_{L_{n-2}}(x_i,x_z)$.

Therefore, ${PQO}_{L_{n-2}}$ satisfies $\left(PQ{O}_{L}3\right)$. In summary, $\forall x_i,x_j\in L$, ${PQO}_{L_{n-2}}$ is a discrete pseudo-quasi overlap function on L. □

Below, we provide some examples of discrete pseudo-quasi overlap functions on L in Theorem 1. Taking $L=\{0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1\},n=9$.

Note that the bolded parts in Table 1, Table 2 and Table 3 represent the values corresponding to $x_i,x_j\in [x_k,x_{k+3}]$ and ${PQO}_{L_k}(x_i,x_j)={PQO}_{L_k^{*}}(x_i,x_j)(k=0,1,2,...,n-2)$ in Theorem 1.

Table 1. A discrete pseudo-quasi overlap function ${PQO}_{L_0}$ on L constructed by the scenario (i) of Theorem 1. ($k=0,x_w=x_3=0.3$).

L	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
0	0	0	0	0	0	0	0	0	0	0	0
0.1	0	0.1	0.1	0.1	0.3	0.3	0.3	0.3	0.3	0.3	0.3
0.2	0	0.1	0.1	0.2	0.3	0.3	0.3	0.3	0.3	0.3	0.3
0.3	0	0.1	0.2	0.3	0.3	0.3	0.3	0.3	0.3	0.3	0.3
0.4	0	0.1	0.2	0.3	0.4	0.4	0.4	0.4	0.4	0.4	0.4
0.5	0	0.1	0.2	0.3	0.4	0.5	0.5	0.5	0.5	0.5	0.5
0.6	0	0.1	0.2	0.3	0.4	0.5	0.6	0.6	0.6	0.6	0.6
0.7	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.7	0.7	0.7
0.8	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.8	0.8
0.9	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	0.9
1	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1

Table 2. A discrete pseudo-quasi overlap function ${PQO}_{L_k}$ on L constructed by the scenario ($ii$) of Theorem 1. ($k=3,x_t=x_6=0.6$).

L	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
0	0	0	0	0	0	0	0	0	0	0	0
0.1	0	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1
0.2	0	0.1	0.1	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.2
0.3	0	0.1	0.2	0.3	0.3	0.3	0.3	0.6	0.6	0.6	0.6
0.4	0	0.1	0.2	0.3	0.4	0.4	0.4	0.6	0.6	0.6	0.6
0.5	0	0.1	0.2	0.3	0.4	0.4	0.5	0.6	0.6	0.6	0.6
0.6	0	0.1	0.2	0.3	0.4	0.5	0.6	0.6	0.6	0.6	0.6
0.7	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.7	0.7	0.7
0.8	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.8	0.8
0.9	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	0.9
1	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1

Table 3. A discrete pseudo-quasi overlap function ${PQO}_{L_{n-2}}$ on L constructed by the scenario ($iii$) of Theorem 1. $(k=7,\theta =1,x_{{\beta }_1}=x_{k-1}=0.6,x_{{\gamma }_1}=x_k=x_{\lambda }=0.7$).

L	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
0	0	0	0	0	0	0	0	0	0	0	0
0.1	0	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1
0.2	0	0.1	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.2
0.3	0	0.1	0.2	0.3	0.3	0.3	0.3	0.3	0.3	0.3	0.3
0.4	0	0.1	0.2	0.3	0.4	0.4	0.4	0.4	0.4	0.4	0.4
0.5	0	0.1	0.2	0.3	0.4	0.5	0.5	0.5	0.5	0.5	0.5
0.6	0	0.1	0.2	0.3	0.4	0.5	0.6	0.6	0.6	0.6	0.6
0.7	0	0.1	0.2	0.3	0.4	0.5	0.7	0.7	0.7	0.7	0.7
0.8	0	0.1	0.2	0.3	0.4	0.5	0.7	0.7	0.8	0.8	0.8
0.9	0	0.1	0.2	0.3	0.4	0.5	0.7	0.7	0.8	0.9	0.9
1	0	0.1	0.2	0.3	0.4	0.5	0.7	0.7	0.8	0.9	1

Of course, the construction method of the scenario ($iii$) in Theorem 1 also applies to $k=2,3,4,\dots n-3$. The specific details are similar to the scenario ($iii$) of Theorem 1.

Based on the scenario ($iii$) of Theorem 1, we obtain the following conclusion:

Proposition 9. 

Let L be a finite chain, $k\in N,k=n-2,[x_{n-2},x_{n+1}]$ be a sub-chain of L, and ${PQO}_{L_{n-2}^{*}}:{[n-2,n+1]}^2\to [n-2,n+1]$ be a discrete pseudo-quasi overlap function on $L_{n-2}^{*}$. Then, $\forall x_i,x_j\in L$, the function ${PQO}_{L_{n-2}^{\left(O\right)}}(O=1,2):L^2\to L$ is defined as follows:

$${PQO}_{L_{n-2}^{\left(O\right)}}(x_i,x_j)=\left\{\begin{array}{cc}{PQO}_{L_{n-2}^{*}}(x_i,x_j),\hfill & if{x}_i,x_j\in [x_{n-2},x_{n+1}]\hfill \\ \min \{{\alpha }_1\left(x_i\right),{\alpha }_1\left(x_j\right)\}.\hfill & otherwise\hfill \end{array}\right.$$

is a discrete pseudo-quasi overlap function on L, among them, ${\alpha }_1\left(x_i\right):L\to L$ and ${\alpha }_2\left(x_j\right):L\to L$, we have the following two different construction forms:

$$\left(i\right)n\ge 5,{\alpha }_1^{\left(5\right)}\left(x_i\right):L\to L,{\alpha }_2^{\left(5\right)}\left(x_j\right):L\to L,\mathit{separately},\mathit{are}\mathit{defined}\mathit{as}\mathit{follows}:$$

$${\alpha }_1^{\left(5\right)}\left(x_i\right)=\left\{\begin{array}{cc}{PQO}_{L_n^{*}}(x_i,x_{n+1}),\hfill & if{x}_i\in [x_{n-2},x_{n+1}]\hfill \\ x_0\hfill & if{x}_i=x_0\hfill \\ x_{\xi }.\hfill & otherwise\hfill \end{array}\right.$$

$${\alpha }_2^{\left(5\right)}\left(x_j\right)=\left\{\begin{array}{cc}{x}_{\pi },\hfill & if{x}_j\in [x_{n-2},x_{n+1}]\hfill \\ x_j\hfill & otherwise\hfill \end{array}\right.$$

where $\xi ,\pi \in N^{+},n-3\le \xi \le n-2,n-3\le \pi \le n+1$, such as $x_{\xi }\le x_{\pi }$.

$$\left(ii\right)n\ge 4,{\alpha }_1^{\left(6\right)}\left(x_i\right):L\to L,{\alpha }_2^{\left(6\right)}\left(x_j\right):L\to L,\mathit{separately},\mathit{are}\mathit{defined}\mathit{as}\mathit{follows}:$$

$${\alpha }_1^{\left(6\right)}\left(x_i\right)=\left\{\begin{array}{cc}{PQO}_{L_k^{*}}(x_i,x_{n+1}),\hfill & if{x}_i\in [x_{n-2},x_{n+1}]\hfill \\ x_{n-2}\hfill & if{x}_i=x_{n-3}\hfill \\ x_{n-3}\hfill & if{x}_i=x_{n-4}\hfill \\ \dots \hfill & \dots \hfill \\ x_3\hfill & if{x}_i=x_2\hfill \\ x_2\hfill & if{x}_i=x_1\hfill \\ x_i\hfill & otherwise\hfill \end{array}\right.$$

$${\alpha }_2^{\left(6\right)}\left(x_j\right)=\left\{\begin{array}{cc}{x}_{\epsilon },\hfill & if{x}_j\in [x_{n-2},x_{n+1}]\hfill \\ x_j\hfill & otherwise\hfill \end{array}\right.$$

where $\epsilon \in N^{+},n-3\le \epsilon \le n+1,$, such as $x_i\le x_{\epsilon }$.

Proof. 

The proof is analogous to Theorem 1. □

Below, we provide some examples of discrete pseudo-quasi overlap functions on L in Proposition 9. Taking $L=\{0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1\},n=9,x_{\xi }=x_{\pi }=0.6.$

Note that the bolded parts in Table 4 and Table 5 represent the values corresponding to $x_i,x_j\in [x_{n-2},x_{n+1}]$ and ${PQO}_{L_{n-2}^{\left(O\right)}}(x_i,x_j)={PQO}_{L_{n-2}^{*}}(x_i,x_j)(k=n-2\in N)$ in Proposition 9.

Table 4. A discrete pseudo-quasi overlap function ${PQO}_{L_{n-2}^{\left(1\right)}}$ on L constructed by the scenario (i) of Proposition 9. ($k=7,x_{\xi }=x_{\pi }=0.6$).

L	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
0	0	0	0	0	0	0	0	0	0	0	0
0.1	0	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1
0.2	0	0.1	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.2
0.3	0	0.1	0.2	0.3	0.3	0.3	0.3	0.3	0.3	0.3	0.3
0.4	0	0.1	0.2	0.3	0.4	0.4	0.4	0.4	0.4	0.4	0.4
0.5	0	0.1	0.2	0.3	0.4	0.5	0.5	0.5	0.5	0.5	0.5
0.6	0	0.1	0.2	0.3	0.4	0.5	0.6	0.6	0.6	0.6	0.6
0.7	0	0.6	0.6	0.6	0.6	0.6	0.6	0.7	0.7	0.7	0.7
0.8	0	0.6	0.6	0.6	0.6	0.6	0.6	0.7	0.8	0.8	0.8
0.9	0	0.6	0.6	0.6	0.6	0.6	0.6	0.7	0.8	0.9	0.9
1	0	0.6	0.6	0.6	0.6	0.6	0.6	0.7	0.8	0.9	1

Table 5. A discrete pseudo-quasi overlap function ${PQO}_{L_{n-2}^{\left(2\right)}}$ on L constructed by the scenario ($ii$) of Proposition 9. ($k=7,x_{\epsilon }=0.7$).

L	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
0	0	0	0	0	0	0	0	0	0	0	0
0.1	0	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1
0.2	0	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.2	0.2
0.3	0	0.2	0.3	0.3	0.3	0.3	0.3	0.3	0.3	0.3	0.3
0.4	0	0.2	0.3	0.4	0.4	0.4	0.4	0.4	0.4	0.4	0.4
0.5	0	0.2	0.3	0.4	0.5	0.5	0.5	0.5	0.5	0.5	0.5
0.6	0	0.2	0.3	0.4	0.5	0.6	0.6	0.6	0.6	0.6	0.6
0.7	0	0.2	0.3	0.4	0.5	0.6	0.7	0.7	0.7	0.7	0.7
0.8	0	0.2	0.3	0.4	0.5	0.6	0.7	0.7	0.8	0.8	0.8
0.9	0	0.2	0.3	0.4	0.5	0.6	0.7	0.7	0.8	0.9	0.9
1	0	0.2	0.3	0.4	0.5	0.6	0.7	0.7	0.8	0.9	1

Likewise, the construction method of Proposition 9 also applies to $k=2,3,4,\dots n-3$. The specific details are similar to Proposition 9.

In summary, the biggest difference between the method of constructing discrete pseudo-quasi overlap functions on finite chains described above and the method of creating quasi-overlap functions on finite chains through ordinal sum in [40] lies in the uniformity of the outcomes. Quasi-overlap functions on finite chains constructed from different sub-chains are the same, whereas the discrete pseudo-quasi overlap functions on finite chains constructed from pseudo-quasi overlap functions on different sub-chains are different.

5. The Application of Discrete Pseudo-Quasi Overlap Functions in Fuzzy Multi-Attribute Group Decision-Making

In this section, we extend the binary discrete pseudo-quasi overlap function on L in Definition 6 to an n-ary discrete pseudo-quasi overlap function on L. Then, we construct an n-dimensional discrete pseudo-quasi overlap function using pseudo-overlap functions, pseudo-quasi-isomorphisms, and integral functions. Furthermore, we apply the n-ary discrete pseudo-quasi overlap function on L to fuzzy multi-attribute group decision-making.

5.1. N-Ary Discrete Pseudo-Quasi Overlap Functions

To start with, we present the concept of n-ary discrete pseudo-quasi overlap functions on L.

Definition 11. 

Let $L=\{0,x_1,x_2,\dots ,x_n,1\}$ be a finite chain. A function ${PQO}_L^n:L^n\to L$ is called an n-ary discrete pseudo-quasi overlap function on L when it satisfies $\forall x_{i+1},x_{i+2},x_{i+3}\dots ,$ $x_{i+n}\in L$,

$\left({PQO}_L^{n}1\right)$ 

${PQO}_L^n(x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n})=x_0\iff$ for $j\in N^{+},1\le j\le n$, such as ${\displaystyle \prod _{j=1}^n}{x}_{i+j}=x_0$;

$\left({PQO}_L^{n}2\right)$ 

${PQO}_L^n(x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n})=x_{n+1}\iff$ for $j\in N^{+},1\le j\le n$, such as ${\displaystyle \prod _{j=1}^n}{x}_{i+j}=x_{n+1}$;

$\left({PQO}_L^{n}3\right)$ 

${PQO}_L^n$ is non-decreasing.

Note that we extend the finite chain L in Definition 11 to [0, 1], then $PQ{O}_L^n$ is an n-ary pseudo-quasi overlap function. Moreover, we can readily provide the definition of n-ary pseudo-quasi overlap functions $PQ{O}^n$, along with its corresponding properties $\left(PQ{O}^{n}1\right),\left(PQ{O}^{n}2\right)$, and $\left(PQ{O}^{n}3\right)$. The definition of n-ary pseudo-quasi overlap functions is similar to Definition 11, so it is omitted here.

Additionally, we assume that $L=$ $\{0,0.001,0.002,\dots ,0.999,1\}$ in Definition 11. In this section, we use $L^{*}$ to represent the finite chain $\{0,0.001,0.002,\dots ,0.999,1\}$.

Example 2. 

Let $L^{*}$ be a finite chain.

$\left(1\right)$ $\forall x_{i+1},x_{i+2},x_{i+3},\dots ,x_{i+n}\in L^{*}$, the function $PQ{O}_{L^{*}}^n:{L^{*}}^n\to L^{*}$,

$$PQ{O}_{L^{*}}^n(x_{i+1},x_{i+2},x_{i+3},\dots ,x_{i+n})=\min (x_{i+1},x_{i+2},x_{i+3},\dots ,x_{i+n-1},{\displaystyle \frac{\left[100x_{i+n}\right]}{100}})$$

where $[·]$ is an integral function, is an n-ary discrete pseudo-quasi overlap function on $L^{*}$.

$\left(2\right)$ $\forall x_{i+1},x_{i+2},x_{i+3},\dots ,x_{i+n}\in L^{*},x_1,x_e,x_f\in L^{*},x_e\ne x_f,x_1<\min \{x_e,x_f\}$, the function $PQ{O}_{L^{*}}^n:{L^{*}}^n\to L^{*}$,

$$\begin{array}{c}{PQO}_{L^{*}}^n(x_{i+1},x_{i+2},x_{i+3},\dots ,x_{i+n})\\ =\left\{\begin{array}{cc}{x}_1,\hfill & if{x}_0<x_1<x_e,x_0<x_{n+1}<x_f\hfill \\ \min (x_{i+1},x_{i+2},x_{i+3},\dots ,x_{i+n}).\hfill & otherwise\hfill \end{array}\right.\end{array}$$

is an n-ary discrete pseudo-quasi overlap function on $L^{*}$.

5.2. Generation of N-Ary Discrete Pseudo-Quasi Overlap Functions

As stated in reference [18], the generation of pseudo-overlap functions comes from n-dimensional overlap functions and a set of weights. Therefore, we construct an n-dimensional discrete pseudo-quasi overlap function based on the pseudo-overlap functions mentioned above, pseudo-quasi-isomorphisms, and integral functions. Below, we introduce the concept of pseudo-quasi-isomorphisms:

Definition 12. 

A unary function $H:[0,1]\to [0,1]$ is called a pseudo-quasi-automorphism when it satisfies $\forall x\in [0,1]$,

$\left(H1\right)$

H is non-decreasing;

$\left(H2\right)$ 

$x=1$ when and only when $H\left(x\right)=1$;

$\left(H3\right)$ 

$x=0$ when and only when $H\left(x\right)=0$.

Obviously, each pseudo-automorphism is a pseudo-quasi-automorphism given in [26]. Conversely, a continuous pseudo-quasi-automorphism is a pseudo-automorphism.

Theorem 2. 

Let $L^{*}$ be a finite chain, $H:\left[0.1\right]\to [0,1]$ be a pseudo-quasi-automorphism, and ${PO}^n:{[0,1]}^n\to [0,1]$ be an n-ary pseudo-overlap function. Then, $\forall x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n}\in [0,1]$, the function ${PQO}^n:{[0,1]}^n\to L^{*}$ is defined as follows:

$$\begin{array}{cc}\hfill {PQO}^n(x_{i+1},x_{i+2},x_{i+3},\dots ,x_{i+n})& =H\left(P{Q}^n(x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n})\right)\hfill \\ \hfill & ={\displaystyle \frac{\left[1000P{Q}^n(x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n})\right]}{1000}}\hfill \end{array}$$

(1)

where $H\left(x\right)={\displaystyle \frac{1}{1000}}F\left(1000x\right),F$ is an integral function, i.e., $F\left(x\right)=\left[x\right]$, and is an n-ary pseudo-quasi overlap function.

Proof. 

Suppose that ${PO}^n$ is an n-ary pseudo-overlap function. $\left(PQ{O}^{n}1\right)$ (Necessity) If we have the following:

$$PQ{O}^n(x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n})=0,$$

then $H\left(P{Q}^n(x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n})\right)=0$, i.e., $P{Q}^n(x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n})=0$. So, for $j\in N^{+},1\le j\le n$, such as ${\displaystyle \prod _{i=1}^n}{x}_{i+j}=0$. (Sufficiency) If $j\in N^{+},1\le j\le n$, ${\displaystyle \prod _{i=1}^n}{x}_{i+j}=0$, then $P{Q}^n(x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n})=0$, that is, we have the following:

$$PQ{O}^n(x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n})=H(P{Q}^n(x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n})=H\left(0\right)=0.$$

Thus, $PQ{O}^n$ satisfies $\left(PQ{O}^{n}1\right)$.$\left(PQ{O}^{n}2\right)$(Necessity) If $PQ{O}^n(x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n})=1$, then $H\left(P{Q}^n(x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n})\right)=1$, that is, $P{Q}^n(x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n})=1$, So, for $j\in N^{+},1\le j\le n$, such as ${\displaystyle \prod _{j=1}^n}{x}_{i+j}=1$. (Sufficiency) If $j\in N^{+},1\le j\le n$, ${\displaystyle \prod _{j=1}^n}{x}_{i+j}=1$, then $P{Q}^n(x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n})=1$, that is, we have the following:

$$PQ{O}^n(x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n})=H\left(P{Q}^n(x_{i+1},x_{i+2},x_{i+3}\dots ,x_{i+n})\right)=H\left(1\right)=1.$$

Thus, $PQ{O}^n$ satisfies $\left(PQ{O}^{n}2\right)$. $\left(PQ{O}^{n}3\right)$ Since F is increasing, H is increasing, and $P{Q}^n$ is increasing, we clearly know that $PQ{O}^n$ is increasing. Thus, $PQ{O}^n$ satisfies $\left(PQ{O}^{n}3\right)$. Therefore, $PQ{O}^n$ is an n-ary pseudo-quasi overlap function. □

Below, we transform the pseudo-quasi overlap function on $[0,1]$ in Theorem 2 into a discrete pseudo-quasi overlap function on $L^{*}$. According to the description of function restrictions in [50] and Theorem 2, we obtain the following conclusion:

Lemma 2. 

Let $L^{*}$ be a finite chain, ${PQO}^n:{[0,1]}^n\to L^{*}$,

$$\begin{array}{cc}\hfill {PQO}^n(x_{i+1},x_{i+2},x_{i+3},\dots ,x_{i+n})& =H\left(P{Q}^n(x_{i+1},x_{i+2},x_{i+3},\dots ,x_{i+n})\right)\hfill \\ \hfill & ={\displaystyle \frac{\left[1000P{Q}^n(x_{i+1},x_{i+2},x_{i+3},\dots ,x_{i+n})\right]}{1000}}\hfill \end{array}$$

(2)

and be an n-ary pseudo-quasi overlap function, and let $L^{*}$ be a subset of $[0,1]$. Then, $\forall x_{i+1},x_{i+2},x_{i+3},\dots ,x_{i+n}\in L^{*}$, the n-ary function ${PQO}^n:{L^{*}}^n\to L^{*}$ is an n-ary discrete pseudo-quasi overlap function on $L^{*}$. Specifically, we use $PQ{O}_{L^{*}}^n$ to represent an n-ary discrete pseudo-quasi overlap function on $L^{*}$.

Based on Theorem 2, Lemma 2, and Examples 6 and 7 of [18], we can obtain the following example:

Example 3. 

Let $L^{*}$ be a finite chain, and $w_1=(0.3,0.3,0.4)$ be a positive weighted vector; the following $PQ{O}_{L_{w_1}^{*}}^{\left(e\right)}(e=1,2,3,4,5):{L^{*}}^3\to L^{*}$ are ternary discrete pseudo-quasi overlap functions on $L^{*}$ generated by $w_1$.

$\left(1\right)$ $\forall x_{i+1},x_{i+2},x_{i+3}\in L^{*}$, the function $PQ{O}_{L_{w_1}^{*}}^{\left(1\right)}:{L^{*}}^3\to L^{*}$,

$$PQ{O}_{L_{w_1}^{*}}^{\left(1\right)}(x_{i+1},x_{i+2},x_{i+3})=\left\{\begin{array}{cc}{x}_0,\hfill & if{x}_{i+1}=x_{i+2}=x_{i+3}=x_0\hfill \\ {\displaystyle \frac{\left[\frac{1000x_{i+1}{x}_{i+2}{x}_{i+3}}{0.3x_{i+1}+0.3x_{i+2}+0.4x_{i+3}}\right]}{1000}}.\hfill & otherwise\hfill \end{array}\right.$$

is a discrete pseudo-quasi overlap function on $L^{*}$.

$\left(2\right)$ $\forall x_{i+1},x_{i+2},x_{i+3}\in L^{*}$, the function $PQ{O}_{L_{w_1}^{*}}^{\left(2\right)}:{L^{*}}^3\to L^{*}$,

$$PQ{O}_{L_{w_1}^{*}}^{\left(2\right)}(x_{i+1},x_{i+2},x_{i+3})={\displaystyle \frac{\left[1000x_{i+1}{x}_{i+2}{x}_{i+3}(0.3x_{i+1}+0.3x_{i+2}+0.4x_{i+3})\right]}{1000}}$$

is a discrete pseudo-quasi overlap function on $L^{*}$.

$\left(3\right)$ $\forall x_{i+1},x_{i+2},x_{i+3}\in L^{*}$, the function $PQ{O}_{L_{w_1}^{*}}^{\left(3\right)}:{L^{*}}^3\to L^{*}$,

$$PQ{O}_{L_{w_1}^{*}}^{\left(3\right)}(x_{i+1},x_{i+2},x_{i+3})=\left\{\begin{array}{cc}{x}_0,\hfill & if{x}_{i+1}=x_{i+2}=x_{i+3}=x_0\hfill \\ {\displaystyle \frac{\left[\frac{330x_{i+1}{x}_{i+2}{x}_{i+3}}{0.12x_{i+2}{x}_{i+3}+0.12x_{i+1}{x}_{i+3}+0.09x_{i+1}{x}_{i+2}}\right]}{1000}}.\hfill & otherwise\hfill \end{array}\right.$$

is a discrete pseudo-quasi overlap function on $L^{*}$.

$\left(4\right)$ $\forall x_{i+1},x_{i+2},x_{i+3}\in L^{*}$, the function $PQ{O}_{L_{w_1}^{*}}^{\left(4\right)}:{L^{*}}^3\to L^{*}$,

$$PQ{O}_{L_{w_1}^{*}}^{\left(4\right)}(x_{i+1},x_{i+2},x_{i+3})={\displaystyle \frac{1}{1000}}\left[{\displaystyle \frac{1000x_{\left(\overline{i+1}\right)}^2{x}_{\left(\overline{i+2}\right)}^4{x}_{\left(\overline{i+3}\right)}^6}{0.4^{2}0.3^{10}}}\right]$$

where $(x_{\left(\overline{i+1}\right)},x_{\left(\overline{i+2}\right)},x_{\left(\overline{i+3}\right)})$ is a permutation of $(0.3x_{i+1},0.3x_{i+2},0.4x_{i+3})$, and it fulfills $x_{\left(\overline{i+3}\right)}\le x_{\left(\overline{i+2}\right)}\le x_{\left(\overline{i+1}\right)}$, is a discrete pseudo-quasi overlap function on $L^{*}$.

$\left(5\right)$ $\forall x_{i+1},x_{i+2},x_{i+3}\in L$, the function $PQ{O}_{L_{w_1}^{*}}^{\left(5\right)}:{L^{*}}^3\to L^{*}$,

$$PQ{O}_{L_{w_1}^{*}}^{\left(5\right)}(x_{i+1},x_{i+2},x_{i+3})={\displaystyle \frac{1}{1000}}\left[{\displaystyle \frac{1000\sqrt{x_{\left(\overline{i+1}\right)}}\sqrt[4]{x_{\left(\overline{i+2}\right)}}\sqrt[6]{x_{\left(\overline{i+3}\right)}}}{\sqrt[4]{0.3^3}\sqrt[6]{0.4}}}\right]$$

where $(x_{\left(\overline{i+1}\right)},x_{\left(\overline{i+2}\right)},x_{\left(\overline{i+3}\right)})$ is a permutation of $(0.3x_{i+1},0.3x_{i+2},0.4x_{i+3})$, and it fulfills $x_{\left(\overline{i+3}\right)}\le x_{\left(\overline{i+2}\right)}\le x_{\left(\overline{i+1}\right)}$, is a discrete pseudo-quasi overlap function on $L^{*}$.

Example 4. 

Let $L^{*}$ be a finite chain, and $w_2=(0.1,0.1,0.2,0.2,0.2,0.2)$ be a positive weighted vector. The following $PQ{O}_{L_{w_2}^{*}}^{\left(e\right)}(e=1,2,3,4,5):{L^{*}}^6\to L^{*}$ are six-variable discrete pseudo-quasi overlap functions on $L^{*}$ generated by $w_2$.

$\left(1\right)$ $\forall x_{i+1},x_{i+2},x_{i+3},x_{i+4},x_{i+5},x_{i+6}\in L^{*}$, the function $PQ{O}_{L_{w_2}^{*}}^{\left(1\right)}:{L^{*}}^6\to L^{*}$,

$$\begin{array}{c}PQ{Q}_{L_{w_2}^{*}}^{\left(1\right)}(x_{i+1},x_{i+2},x_{i+3},x_{i+4},x_{i+5},x_{i+6})\\ =\left\{\begin{array}{cc}{x}_0,\hfill & if{x}_{i+1}=x_{i+2}=x_{i+3}=x_{i+4}\hfill \\ & =x_{i+5}=x_{i+6}=x_0\hfill \\ {\displaystyle \frac{\left[1000\sqrt{\frac{x_{i+1}{x}_{i+2}{x}_{i+3}{x}_{i+4}{x}_{i+5}{x}_{i+6}}{0.1x_{i+1}+0.1x_{i+2}+0.2x_{i+3}+0.2x_{i+4}+0.2x_{i+5}+0.2x_{i+6}}}\right]}{1000}}.\hfill & otherwise\hfill \end{array}\right.\end{array}$$

is a discrete pseudo-quasi overlap function on $L^{*}$.

$\left(2\right)$ $\forall x_{i+1},x_{i+2},x_{i+3},x_{i+4},x_{i+5},x_{i+6}\in L^{*}$, the function $PQ{O}_{L_{w_2}^{*}}^{\left(2\right)}:{L^{*}}^6\to L^{*}$,

$$\begin{array}{c}PQ{O}_{L_{w_2}^{*}}^{\left(2\right)}(x_{i+1},x_{i+2},x_{i+3},x_{i+4},x_{i+5},x_{i+6})\\ ={\displaystyle \frac{\left[1000x_{i+1}{x}_{i+2}{x}_{i+3}{x}_{i+4}{x}_{i+5}{x}_{i+6}(0.1x_{i+1}+0.1x_{i+2}+0.2x_{i+3}+0.2x_{i+4}+0.2x_{i+5}+0.2x_{i+6})\right]}{1000}}\end{array}$$

is a discrete pseudo-quasi overlap function on $L^{*}$.

$\left(3\right)$ $\forall x_{i+1},x_{i+2},x_{i+3},x_{i+4},x_{i+5},x_{i+6}\in L$, the function $PQ{O}_{L_{w_2}}^{\left(3\right)}:{L^{*}}^6\to L^{*}$,

$$\begin{array}{c}PQ{O}_{L_{w_2}^{*}}^{\left(3\right)}(x_{i+1},x_{i+2},x_{i+3},x_{i+4},x_{i+5},x_{i+6})\\ =\left\{\begin{array}{cc}{x}_0,\hfill & if{x}_{i+1}{x}_{i+2}{x}_{i+3}{x}_{i+4}{x}_{i+5}{x}_{i+6}=x_0\hfill \\ {\displaystyle \frac{\left[\frac{8000}{\frac{2}{x_{i+1}}+\frac{2}{x_{i+2}}+\frac{1}{x_{i+3}}+\frac{1}{x_{i+4}}+\frac{1}{x_{i+5}}+\frac{1}{x_{i+6}}}\right]}{1000}}.\hfill & otherwise\hfill \end{array}\right.\end{array}$$

is a discrete discrete pseudo-quasi overlap function on $L^{*}$.

$\left(4\right)$ $\forall x_{i+1},x_{i+2},x_{i+3},x_{i+4},x_{i+5},x_{i+6}\in L$, the function $PQ{O}_{L_{w_2}^{*}}^{\left(4\right)}:{L^{*}}^6\to L^{*}$,

$$PQ{O}_{L_w^{*}}^{\left(4\right)}(x_{i+1},x_{i+2},x_{i+3},x_{i+4},x_{i+5},x_{i+6})={\displaystyle \frac{\left[\frac{1000x_{\left(\overline{i+1}\right)}^2{x}_{\left(\overline{i+2}\right)}^4{x}_{\left(\overline{i+3}\right)}^6{x}_{\left(\overline{i+4}\right)}^8{x}_{\left(\overline{i+5}\right)}^{10}{x}_{\left(\overline{i+6}\right)}^{12}}{0.2^{20}{0.1}^{22}}\right]}{1000}}$$

where $(x_{\left(\overline{i+1}\right)},x_{\left(\overline{i+2}\right)},x_{\left(\overline{i+3}\right)},x_{\left(\overline{i+4}\right)},x_{\left(\overline{i+5}\right)},x_{\left(\overline{i+6}\right)})$ is a permutation of $(0.1x_{i+1},0.1x_{i+2},0.2x_{i+3},0.2x_{i+4},0.2x_{i+5},0.2x_{i+6})$, and it fulfills $x_{\left(\overline{i+6}\right)}\le x_{\left(\overline{i+5}\right)}\le x_{\left(\overline{i+4}\right)}\le x_{\left(\overline{i+3}\right)}\le x_{\left(\overline{i+2}\right)}\le x_{\left(\overline{i+1}\right)}$, is a discrete pseudo-quasi overlap function on $L^{*}$.

$\left(5\right)$ $\forall x_{i+1},x_{i+2},x_{i+3},x_{i+4},x_{i+5},x_{i+6}\in L^{*}$, the function $PQ{O}_{L_{w_2}^{*}}^{\left(5\right)}:{L^{*}}^6\to L^{*}$,

$$PQ{O}_{L_w^{*}}^{\left(5\right)}(x_{i+1},x_{i+2},x_{i+3},x_{i+4},x_{i+5},x_{i+6})={\displaystyle \frac{1}{1000}}\left[{\displaystyle \frac{1000\sqrt{x_{\left(\overline{i+1}\right)}}\sqrt[4]{x_{\left(\overline{i+2}\right)}}\sqrt[6]{x_{\left(\overline{i+3}\right)}}\sqrt[8]{x_{\left(\overline{i+4}\right)}}\sqrt[10]{x_{\left(\overline{i+5}\right)}}\sqrt[12]{x_{\left(\overline{i+6}\right)}}}{\sqrt[4]{0.1^3}\sqrt[40]{0.2^{19}}}}\right]$$

where $(x_{\left(\overline{i+1}\right)},x_{\left(\overline{i+2}\right)},x_{\left(\overline{i+3}\right)},x_{\left(\overline{i+4}\right)},x_{\left(\overline{i+5}\right)},x_{\left(\overline{i+6}\right)})$ is a permutation of $(0.1x_{i+1},0.1x_{i+2},0.2x_{i+3},0.2x_{i+4},0.2x_{i+5},0.2x_{i+6})$, and it fulfills $x_{\left(\overline{i+1}\right)}\le x_{\left(\overline{i+2}\right)}\le x_{\left(\overline{i+3}\right)}\le x_{\left(\overline{i+4}\right)}\le x_{\left(\overline{i+5}\right)}\le x_{\left(\overline{i+6}\right)}$, is a discrete pseudo-quasi overlap function on $L^{*}$.

Next, we apply the discrete pseudo-quasi overlap functions on $L^{*}$ proposed above to fuzzy multi-attribute group decision-making.

5.3. An Application of Discrete Pseudo-Quasi Overlap Functions in Fuzzy Multi-Attribute Group Decision-Making

At present, the aggregation functions used in most applications of fuzzy multi-attribute group decision-making are continuous, such as the Sugeno integral based on overlap functions in [47], the overlap function in [49], and the pseudo-overlap function in [18]. However, in practical applications of fuzzy multi-attribute group decision-making, the data objects involved are generally discrete. Therefore, we apply the n-ary discrete pseudo-quasi overlap function constructed above to fuzzy multi-attribute group decision-making. Firstly, we briefly introduce the concept of fuzzy multi-attribute group decision-making.

A solution to the fuzzy multi-attribute group decision-making problem (FMAGDMP) involves selecting the most favorable options from a list of alternatives, taking into account various attributes of the alternatives as well as the perspectives of the specialist group.

Generally, in a FMAGDMP, let $A=\{a_1,a_2,\dots ,a_n\}$ be a discrete finite set of feasible alternatives, $U=\{u_1,u_2,\dots ,u_m\}$ be a set of attributes, $E=\{{\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_k\}$ be a set of decision makers, and $w_1={(w_{11},w_{12},\dots ,w_{1k})}^T,w_2={(w_{21},w_{22},\dots ,w_{2m})}^T$ be a positive weighted vector. Each decision maker ${\epsilon }_t$ creates a decision matrix $S^t={\left({s_{ij}}^{\left(t\right)}\right)}_{n\times m}$, with the columns denoting the attributes and the rows indicating the feasible alternatives. In traditional decision-making, if the feasible alternative $a_i$ has the attribute $u_j$, then the decision makers ${\epsilon }_t$ believe that the position ${s_{ij}}^{\left(t\right)}$ of $S^{\left(t\right)}$ has the value of 1, and if not, the position ${s_{ij}}^{\left(t\right)}$ of $S^{\left(t\right)}$ has the value of 0. However, under certain circumstances, some features are usually vague, such as “reasonable price”, “market depression”, and “currency inflation”, which are essentially ambiguous. Therefore, we need to treat them as fuzzy sets. In this instance, the value at position ${s_{ij}}^{\left(t\right)}$ represents the membership degree, that is, a value in [0, 1], of the alternative $a_i$ to the fuzzy set connected to the attribute $u_j$. Generally speaking, profit and expenses are two important attributes. For instance, while “risk of investment” is an expense attribute, “quality of construction project” is a profit attribute. In addition, we assume that $\phi$ is an index set of the profit attributes.

We provide the solution to FMAGDMP as follows:

• Step 1. Use the following Formula $\left(3\right)$ to convert each decision matrix $S^{\left(t\right)}={\left({s_{ij}}^{\left(t\right)}\right)}_{n\times m}$ into a standard decision matrix $N^{\left(t\right)}={\left({n_{ij}}^{\left(t\right)}\right)}_{n\times m}$;

  $${n_{ij}}^{\left(t\right)}=\left\{\begin{array}{cc}{s_{ij}}^{\left(t\right)},\hfill & ifj\in \phi \hfill \\ 1-{s_{ij}}^{\left(t\right)}.\hfill & otherwise\hfill \end{array}\right.$$

  (3)
• Step 2. Generate a congregate decision matrix $Q={\left(q_{ij}\right)}_{n\times m}$ by aggregating the standard decision matrix $N^{\left(t\right)}={\left({n_{ij}}^{\left(t\right)}\right)}_{n\times m}$ according to an n-dimensional discrete pseudo-quasi overlap function $PQ{O}_{L_{w_1}^{*}}^{\left(e\right)}(e\in N^{+},1\le e\le 5)$ on $L^{*}$, where the aggregation method is shown in Formula $\left(4\right)$ below;

  $$q_{ij}=PQ{O}_{L_{w_1}^{*}}^{\left(e\right)}({n_{ij}}^{\left(1\right)},{n_{ij}}^{\left(2\right)},\dots ,{n_{ij}}^{\left(k\right)})$$

  (4)
• Step 3. Determine the total preference vector $tp{v}_i$ for each alternative $a_i$ by aggregating the membership degrees to each attribute $u_j$ using $PQ{O}_{L_{w_2}^{*}}^{\left(e\right)}(e\in N^{+},1\le f\le 5)$ on $L^{*}$; Formula (5) below shows the aggregating approach:

  $$tp{v}_i=PQ{O}_{L_{w_2}}^{\left(e\right)}(q_{i1},q_{i2},\dots ,q_{im})$$

  (5)
• Step 4. Sort the alternatives based on the overall preference values in descending order and select the alternative with the highest value.

Next, we demonstrate the application of the above method through the example given in [43]. We assume that investors plan to contribute a portion of their funds to an enterprise. Making use of a market analysis, investors narrow down the range of potential enterprises to six:

$a_1:$ a chemical enterprise;

$a_2:$ a food firm;

$a_3:$ a computer corporation;

$a_4:$ an automobile firm;

$a_5:$ a furniture corporation;

$a_6:$ a pharmaceutical enterprise.

Three specialists or decision makers $({\epsilon }_1,{\epsilon }_2,{\epsilon }_3)$ with corresponding weight vectors $w_1=(0.3,0.3,0.4)$ assist the investor.

Six attributes are established by the specialist panel to assess the investments.

The profit attributes include the following:

$u_1:$ profits in the immediate term;

$u_2:$ profits in the medium term;

$u_3:$ profits over the long haul.

The expense attributes include the following:

$u_4:$ investing in danger;

$u_5:$ investment challenge;

$u_6:$ additional detrimental aspects of investment.

The assessments provided by the specialists regarding the degree to which the investments align with the attributes are shown in Table 6, Table 7 and Table 8, forming the decision matrix for each specialist.

Table 6. Evaluation of specialist ${\epsilon }_1$.

${\mathit{S}}^{\left(1\right)}$	${\mathit{u}}_1$	${\mathit{u}}_2$	${\mathit{u}}_3$	${\mathit{u}}_4$	${\mathit{u}}_5$	${\mathit{u}}_6$
$a_1$	0.7	0.8	0.6	0.7	0.5	0.9
$a_2$	0.8	0.6	0.9	0.7	0.6	0.7
$a_3$	0.5	0.4	0.8	0.3	0.8	0.8
$a_4$	0.6	0.7	0.6	0.7	0.8	0.6
$a_5$	0.9	0.8	0.4	0.7	0.7	0.8
$a_6$	0.8	0.3	0.7	0.7	0.6	0.7

Table 7. Evaluation of specialist ${\epsilon }_2$.

${\mathit{S}}^{\left(2\right)}$	${\mathit{u}}_1$	${\mathit{u}}_2$	${\mathit{u}}_3$	${\mathit{u}}_4$	${\mathit{u}}_5$	${\mathit{u}}_6$
$a_1$	0.6	0.8	0.5	0.6	0.4	0.8
$a_2$	0.7	0.6	0.8	0.6	0.7	0.7
$a_3$	0.7	0.6	0.8	0.7	0.8	0.8
$a_4$	0.6	0.7	0.5	0.6	0.8	0.7
$a_5$	0.7	0.8	0.7	0.7	0.6	0.8
$a_6$	0.6	0.4	0.8	0.7	0.6	0.7

Table 8. Evaluation of specialist ${\epsilon }_3$.

${\mathit{S}}^{\left(3\right)}$	${\mathit{u}}_1$	${\mathit{u}}_2$	${\mathit{u}}_3$	${\mathit{u}}_4$	${\mathit{u}}_5$	${\mathit{u}}_6$
$a_1$	0.7	0.6	0.6	0.6	0.4	0.7
$a_2$	0.7	0.6	0.7	0.6	0.6	0.7
$a_3$	0.6	0.5	0.8	0.5	0.8	0.8
$a_4$	0.6	0.7	0.7	0.5	0.8	0.6
$a_5$	0.7	0.8	0.6	0.7	0.6	0.8
$a_6$	0.4	0.5	0.9	0.7	0.6	0.6

After applying Formula (3) from step 1 to the decision matrices $S^{\left(1\right)},S^{\left(2\right)}$ and $S^{\left(3\right)}$ mentioned above, we obtain the standard decision matrices $N^{\left(1\right)},N^{\left(2\right)}$, and $N^{\left(3\right)}$, which are shown in Table 9, Table 10 and Table 11 in that order.

Table 9. Standardization of specialist ${\epsilon }_1$ decision matrix.

${\mathit{N}}^{\left(1\right)}$	${\mathit{u}}_1$	${\mathit{u}}_2$	${\mathit{u}}_3$	${\mathit{u}}_4$	${\mathit{u}}_5$	${\mathit{u}}_6$
$a_1$	0.7	0.8	0.6	0.3	0.5	0.1
$a_2$	0.8	0.6	0.9	0.3	0.4	0.3
$a_3$	0.5	0.4	0.8	0.7	0.2	0.2
$a_4$	0.6	0.7	0.6	0.3	0.2	0.4
$a_5$	0.9	0.8	0.4	0.3	0.3	0.2
$a_6$	0.8	0.3	0.7	0.3	0.4	0.3

Table 10. Standardization of specialist ${\epsilon }_2$ decision matrix.

${\mathit{N}}^{\left(2\right)}$	${\mathit{u}}_1$	${\mathit{u}}_2$	${\mathit{u}}_3$	${\mathit{u}}_4$	${\mathit{u}}_5$	${\mathit{u}}_6$
$a_1$	0.6	0.8	0.5	0.4	0.6	0.2
$a_2$	0.7	0.6	0.8	0.4	0.3	0.3
$a_3$	0.7	0.6	0.8	0.3	0.2	0.2
$a_4$	0.6	0.7	0.5	0.4	0.2	0.3
$a_5$	0.7	0.8	0.7	0.3	0.4	0.2
$a_6$	0.6	0.4	0.8	0.3	0.4	0.3

Table 11. Standardization of specialist ${\epsilon }_3$ decision matrix.

${\mathit{N}}^{\left(3\right)}$	${\mathit{u}}_1$	${\mathit{u}}_2$	${\mathit{u}}_3$	${\mathit{u}}_4$	${\mathit{u}}_5$	${\mathit{u}}_6$
$a_1$	0.7	0.6	0.6	0.4	0.6	0.3
$a_2$	0.7	0.6	0.7	0.4	0.4	0.3
$a_3$	0.6	0.5	0.8	0.5	0.2	0.2
$a_4$	0.6	0.7	0.7	0.5	0.2	0.4
$a_5$	0.7	0.8	0.6	0.3	0.4	0.2
$a_6$	0.4	0.5	0.9	0.3	0.4	0.4

The congregate decision matrix Q of Table 12 is produced by applying $PQ{O}_{L_{w_1}^{*}}^{\left(1\right)}$ of Example 3 to the standard decision matrices $N^{\left(1\right)},N^{\left(2\right)}$, and $N^{\left(3\right)}$ above.

Table 12. Congregate decision matrix.

Q	${\mathit{u}}_1$	${\mathit{u}}_2$	${\mathit{u}}_3$	${\mathit{u}}_4$	${\mathit{u}}_5$	${\mathit{u}}_6$
$a_1$	0.439	0.533	0.316	0.130	0.316	0.029
$a_2$	0.537	0.360	0.638	0.130	0.130	0.090
$a_3$	0.350	0.240	0.640	0.210	0.040	0.040
$a_4$	0.360	0.490	0.344	0.146	0.040	0.130
$a_5$	0.580	0.640	0.295	0.090	0.130	0.040
$a_6$	0.331	0.146	0.622	0.090	0.160	0.106

Afterward, the total preference vector $tp{v}_i$ is determined by taking into account $PQ{O}_{L_{w_2}^{*}}^{\left(1\right)}$ of Example 4. Table 13 presents the final result. (Specifically, $[·]$ in $PQ{O}_{L_{w_1}^{*}}^{\left(1\right)}$ and $PQ{O}_{L_{w_2}^{*}}^{\left(1\right)}$ only represents the rounding function. For other types of integral functions, such as floor, ceil, and fix, the results obtained are similar to those of the round function.)

Table 13. Total preference vector.

$\mathit{TPV}$	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{a}}_4$	${\mathit{a}}_5$	${\mathit{a}}_6$
$tp{v}_i$	0.018	0.026	0.009	0.015	0.015	0.014

Eventually, we obtain the descending order of the alternative $a_i$ by utilizing Table 13:

$$a_2>a_1>a_4=a_5>a_6>a_3$$

We are aware that the weighted discrete pseudo-quasi overlap functions used in steps 2 and 3 of the FMAGDMP solution are different, and the final ranking of the alternative $a_i$ is dissimilar. In Table 14, we obtain different rankings by utilizing $PQ{O}_{L_{w_1}^{*}}^{\left(e\right)}$ and $PQ{O}_{L_{w_2}^{*}}^{\left(e\right)}(e\in N^{+},e=1,2,3,4,5)$ from Examples 3 and 4 and other aggregation functions in [48,50]. Moreover, the above rankings are generated by different aggregation functions under the same FMAGDMP solution.

Table 14. Ranks obtained by different weighted discrete pseudo-quasi functions and other aggregation functions in [49,51].

Congregate Aggregation Function	Total Aggregation Function	Ranking
$Maximum$	$Maximum$	$a_2>a_4>a_5>a_1>a_3>a_6$
$Minimum$	$Minimum$	$a_3>a_5>a_1>a_4>a_2>a_6$
$W{A}_w^P$	$W{A}_w^P$	$a_2>x_3>a_6>a_4>a_1>a_5$
$W{A}_w^{O_{0.5M}^3}$	$W{A}_w^{O_{0.5M}^3}$	$a_5>a_2>a_1>a_4>a_3>a_6$
$W{A}_w^{O_{2M}^3}$	$W{A}_w^{O_{2M}^3}$	$a_2>a_1>a_5>a_4>a_3>a_6$
$PQ{O}_{L_{w_1}^{*}}^{\left(1\right)}$	$PQ{O}_{L_{w_2}^{*}}^{\left(1\right)}$	$a_2>a_1>a_4=a_5>a_6>a_3$
$PQ{O}_{L_{w_1}^{*}}^{\left(1\right)}$	$PQ{O}_{L_{w_2}^{*}}^{\left(2\right)}$	$a_2=a_1=a_4=a_5=a_6=a_3$
$PQ{O}_{L_{w_1}^{*}}^{\left(1\right)}$	$PQ{O}_{L_{w_2}^{*}}^{\left(3\right)}$	$a_2>a_6>a_4>a_5>a_1>a_3$
$PQ{O}_{L_{w_1}^{*}}^{\left(2\right)}$	$PQ{O}_{L_{w_2}^{*}}^{\left(1\right)}$	$a_2=a_1=a_4=a_5=a_6=a_3$
$PQ{O}_{L_{w_1}^{*}}^{\left(2\right)}$	$PQ{O}_{L_{w_2}^{*}}^{\left(2\right)}$	$a_2=a_1=a_4=a_5=a_6=a_3$
$PQ{O}_{L_{w_1}^{*}}^{\left(2\right)}$	$PQ{O}_{L_{w_2}^{*}}^{\left(3\right)}$	$a_2>a_6>a_4>a_5>a_3=a_1$
$PQ{O}_{L_{w_1}^{*}}^{\left(3\right)}$	$PQ{O}_{L_{w_2}^{*}}^{\left(1\right)}$	$a_2>a_1>a_5>a_4>a_6>a_3$
$PQ{O}_{L_{w_1}^{*}}^{\left(3\right)}$	$PQ{O}_{L_{w_2}^{*}}^{\left(2\right)}$	$a_2>a_1>a_3>a_4=a_5=a_6$
$PQ{O}_{L_{w_1}^{*}}^{\left(3\right)}$	$PQ{O}_{L_{w_2}^{*}}^{\left(3\right)}$	$a_2>a_5>a_4>a_1>a_6>a_3$
$PQ{O}_{L_{w_1}^{*}}^{\left(4\right)}$	$PQ{O}_{L_{w_2}^{*}}^{\left(4\right)}$	$a_2=a_6=a_1=a_4=a_5=a_3$
$PQ{O}_{L_{w_1}^{*}}^{\left(5\right)}$	$PQ{O}_{L_{w_2}^{*}}^{\left(5\right)}$	$a_2>a_5>a_4>a_1>a_6>a_3$

From Table 14, we notice that the eleven aggregation methods generated by $PQ{O}_{L_{w_1}^{*}}^{\left(e\right)}$ and $PQ{O}_{L_{w_2}^{*}}^{\left(e\right)}(e\in N^{+},e=1,2,3,4,5)$ of Examples 3 and 4 resulted in seven different sorts. In addition, among these seven different sorts, all sorts indicate that $a_2$ is the best, while most sorts (five sorts) show that $a_3$ is the worst.

By analyzing Table 14, we can see that the rankings generated by different aggregation functions under the same FMAGDMP solution are slightly different, and compared to other aggregation functions, the rankings obtained using discrete pseudo-quasi overlap functions are more reasonable.

As mentioned above, we use weighted discrete pseudo-quasi overlap functions to fuse information. However, in practical applications, there may be situations without weight vectors. Therefore, we choose other types of discrete pseudo-quasi overlap functions as aggregation functions to solve FMAGDMP. Of course, this type of discrete pseudo-quasi overlap function is significantly different from Examples 3 and 4. It implies the importance of various expert decisions or attributes in the function formula itself. Below, we apply this type of discrete pseudo-quasi overlap function to the previous approach for solving FMAGDMP.

Based on Theorem 2, Lemma 2, and [18], we obtain the following example:

Example 5. 

Let $L^{*}$ be a finite chain; the following $PQ{O}_{L^{*}}^{\left(f\right)}(f=1,2,3):{L^{*}}^3\to L^{*}$ are ternary discrete pseudo-quasi overlap functions on $L^{*}$.

$\left(1\right)$$\forall x_{i+1},x_{i+2},x_{i+3}\in L^{*}$, the function $PQ{O}_{L^{*}}^{\left(1\right)}:{L^{*}}^3\to L^{*}$ is defined as follows:

$$PQ{O}_{L^{*}}^{\left(1\right)}(x_{i+1},x_{i+2},x_{i+3})={\displaystyle \frac{1}{1000}}\left[1000\sqrt[6]{x_{i+1}}\sqrt[4]{x_{i+2}}\sqrt{x_{i+3}}\right]$$

and is a discrete pseudo-quasi overlap function on $L^{*}$.

$\left(2\right)$$\forall x_{i+1},x_{i+2},x_{i+3}\in L^{*}$, the function $PQ{O}_{L^{*}}^{\left(2\right)}:{L^{*}}^3\to L^{*}$ is defined as follows:

$$PQ{O}_{L^{*}}^{\left(2\right)}(x_{i+1},x_{i+2},x_{i+3})=\left\{\begin{array}{cc}{x}_0,\hfill & if{x}_{i+1}+x_{i+2}+x_{i+3}=x_0\hfill \\ {\displaystyle \frac{1}{1000}}\left[{\displaystyle \frac{6000x_{i+1}{x}_{i+2}{x}_{i+3}}{3x_{i+1}+2x_{i+2}+x_{i+3}}}\right].\hfill & otherwise\hfill \end{array}\right.$$

and is a discrete pseudo-quasi overlap function on $L^{*}$.

$\left(3\right)$$\forall x_{i+1},x_{i+2},x_{i+3}\in L^{*}$, the function $PQ{O}_{L^{*}}^{\left(3\right)}:{L^{*}}^3\to L^{*}$ is defined as follows:

$$PQ{O}_{L^{*}}^{\left(3\right)}(x_{i+1},x_{i+2},x_{i+3})={\displaystyle \frac{1}{1000}}\left[{\displaystyle \frac{2000\sqrt[4]{x_{i+1}}\sqrt[3]{x_{i+2}}\sqrt{x_{i+3}}}{1+\sqrt[4]{x_{i+1}}\sqrt[3]{x_{i+2}}\sqrt{x_{i+3}}}}\right]$$

and is a discrete pseudo-quasi overlap function on $L^{*}$.

Example 6. 

Let $L^{*}$ be a finite chain; the following $PQ{O}_{L^{*}}^{\left(f\right)}(f=4,5,6):L^6\to L$ are six-variable discrete pseudo-quasi overlap functions on $L^{*}$.

$\left(1\right)$$\forall x_{i+1},x_{i+2},x_{i+3},x_{i+4},x_{i+5},x_{i+6}\in L^{*}$, the function $PQ{O}_{L^{*}}^{\left(4\right)}:{L^{*}}^6\to L^{*}$,

$$PQ{O}_{L^{*}}^{\left(4\right)}(x_{i+1},x_{i+2},x_{i+3},x_{i+4},x_{i+5},x_{i+6})={\displaystyle \frac{1}{1000}}\left[1000\sqrt[12]{x_{i+1}}\sqrt[10]{x_{i+2}}\sqrt[8]{x_{i+3}}\sqrt[6]{x_{i+4}}\sqrt[4]{x_{i+5}}\sqrt{x_{i+6}}\right]$$

is a discrete pseudo-quasi overlap function on $L^{*}$.

$\left(2\right)$$\forall x_{i+1},x_{i+2},x_{i+3},x_{i+4},x_{i+5},x_{i+6}\in L^{*}$, the function $PQ{O}_{L^{*}}^{\left(5\right)}:{L^{*}}^6\to L^{*}$,

$$\begin{array}{c}PQ{O}_{L^{*}}^{\left(5\right)}(x_{i+1},x_{i+2},x_{i+3},x_{i+4},x_{i+5},x_{i+6})\\ =\left\{\begin{array}{cc}{x}_0,\hfill & if{x}_{i+1}+x_{i+2}+x_{i+3}+x_{i+4}\hfill \\ & +x_{i+5}+x_{i+6}=x_0\hfill \\ {\displaystyle \frac{1}{1000}}\left[{\displaystyle \frac{10000x_{i+1}{x}_{i+2}{x}_{i+3}{x}_{i+4}{x}_{i+5}{x}_{i+6}}{4x_{i+1}+2x_{i+2}+x_{i+3}+x_{i+4}+x_{i+5}+x_{i+6}}}\right].\hfill & otherwise\hfill \end{array}\right.\end{array}$$

is a discrete pseudo-quasi overlap function on $L^{*}$.

$\left(3\right)$$\forall x_{i+1},x_{i+2},x_{i+3},x_{i+4},x_{i+5},x_{i+6}\in L^{*}$, the function $PQ{O}_{L^{*}}^{\left(6\right)}:L^6\to L^{*}$,

$$PQ{O}_{L^{*}}^{\left(6\right)}(x_{i+1},x_{i+2},x_{i+3},x_{i+4},x_{i+5},x_{i+6})={\displaystyle \frac{1}{1000}}\left[{\displaystyle \frac{2000\sqrt[7]{x_{i+1}}\sqrt[6]{x_{i+2}}\sqrt[5]{x_{i+3}}\sqrt[4]{x_{i+4}}\sqrt[3]{x_{i+5}}\sqrt{x_{i+6}}}{1+\sqrt[7]{x_{i+1}}\sqrt[6]{x_{i+2}}\sqrt[5]{x_{i+3}}\sqrt[4]{x_{i+4}}\sqrt[3]{x_{i+5}}\sqrt{x_{i+6}}}}\right]$$

is a discrete pseudo-quasi overlap function on $L^{*}$.

Similar to the previous approach to solving FMAGDMP, we obtain different rankings by means of $PQ{O}_{L^{*}}^{\left(f\right)}(f\in N^{+},f=1,2,3,4,5,6)$ in Examples 5 and 6, as shown in Table 15 below.

Table 15. Ranks obtained by different discrete pseudo-quasi overlap functions.

Congregate Aggregation Function	Total Aggregation Function	Ranking
$PQ{O}_{L^{*}}^{\left(1\right)}$	$PQ{O}_{L^{*}}^{\left(4\right)}$	$a_6>a_2>a_4>a_1>a_5>a_3$
$PQ{O}_{L^{*}}^{\left(1\right)}$	$PQ{O}_{L^{*}}^{\left(5\right)}$	$a_2>a_1>a_6>a_4>a_5>a_3$
$PQ{O}_{L^{*}}^{\left(1\right)}$	$PQ{O}_{L^{*}}^{\left(6\right)}$	$a_2>a_6>a_1>a_4>a_5>a_3$
$PQ{O}_{L^{*}}^{\left(2\right)}$	$PQ{O}_{L^{*}}^{\left(4\right)}$	$a_2>a_6>a_4>a_1>a_5>a_3$
$PQ{O}_{L^{*}}^{\left(2\right)}$	$PQ{O}_{L^{*}}^{\left(5\right)}$	$a_2=a_1=a_4=a_5=a_6=a_3$
$PQ{O}_{L^{*}}^{\left(2\right)}$	$PQ{O}_{L^{*}}^{\left(6\right)}$	$a_2>a_6>a_1>a_4>a_5>a_3$
$PQ{O}_{L^{*}}^{\left(3\right)}$	$PQ{O}_{L^{*}}^{\left(4\right)}$	$a_6>a_2>a_4>a_1>a_5>a_3$
$PQ{O}_{L^{*}}^{\left(3\right)}$	$PQ{O}_{L^{*}}^{\left(5\right)}$	$a_2>a_1>a_6>a_4=a_5>a_3$
$PQ{O}_{L^{*}}^{\left(3\right)}$	$PQ{O}_{L^{*}}^{\left(6\right)}$	$a_2>a_6>a_1>a_4>a_5>a_3$

From Table 15, it can clearly be seen that the nine different aggregation methods created by $PQ{O}_L^{\left(f\right)}(f\in N^{+},f=1,2,3,4,5,6)$ from Examples 5 and 6 bring seven different sorts, and among these sorts, the great majority of sorts (five sorts) consider $a_2$ to be the best, while all sorts consider $a_3$ to be the worst. Moreover, the rankings in Table 14 and Table 15 can be integrated, and further analysis shows that discrete pseudo-quasi overlap functions may be more flexible in fuzzy multi-attribute applications compared to other aggregate functions.

In summary, the discrete pseudo-quasi overlap function applied to fuzzy multi-attribute group decision-making not only aggregates multiple pieces of information but also reflects the significance of different factors, such as the importance of attributes and specialists. More importantly, under the same fuzzy multi-attribute decision-making solution, according to Table 14 and Table 15, and references [18,49,51], we can see that compared to the overlap functions and pseudo-overlap functions, which contain continuity and symmetry and have limitations, the discrete pseudo-quasi overlap function proposed in this paper offers a wider range of applications and greater flexibility.

6. Conclusions

In this paper, we first introduce the concept of discrete pseudo-quasi overlap functions on finite chains and discuss their associated properties. Then, we present pseudo-quasi overlap functions on sub-chains; based on this, we construct discrete pseudo-quasi overlap functions on finite chains through pseudo-quasi overlap functions on sub-chains. Compared to quasi-overlap functions on finite chains constructed using ordinal sums, the discrete pseudo-quasi overlap functions on finite chains derived from pseudo-quasi overlap functions on different sub-chains are not the same. Finally, we present the concept of pseudo-quasi-automorphisms by removing the continuity assumption from pseudo-automorphisms, and we use pseudo-overlap functions, pseudo-quasi-isomorphisms, and integral functions to create discrete pseudo-quasi overlap functions expressed as fractions on finite chains. More importantly, we apply the discrete pseudo-quasi overlap function constructed above to fuzzy multi-attribute group decision-making. The results demonstrate that, compared to overlap functions, pseudo-overlap functions, and other aggregation functions, the proposed approach is both more practical and more flexible.

The results of this paper not only enrich the theoretical research on overlap functions but also provide practical guidance for their application. In future research, we will continue to study the theoretical knowledge and practical applications related to pseudo-quasi overlap functions, which can be divided into the following aspects:

(1) Deriving residual-type implication operators using pseudo-quasi overlap functions and combining them with various inference algorithms;

(2) Extending the pseudo-quasi overlap function to a more general form and studying its related properties;

(3) Exploring the application of the pseudo-quasi overlap function as a relatively broad aggregation function; this can be applied in other fields such as attribute reduction, fuzzy mathematical morphology, and image processing.