Pseudo Overlap Functions, Fuzzy Implications and Pseudo Grouping Functions with Applications

: Overlap and grouping functions are important aggregation operators, especially in information fusion, classiﬁcation and decision-making problems. However, when we do more in-depth application research (for example, non-commutative fuzzy reasoning, complex multi-attribute decision making and image processing), we ﬁnd overlap functions as well as grouping functions are required to be commutative (or symmetric), which limit their wide applications. For the above reasons, this paper expands the original notions of overlap functions and grouping functions, and the new concepts of pseudo overlap functions and pseudo grouping functions are proposed on the basis of removing the commutativity of the original functions. Some examples and construction methods of pseudo overlap functions and pseudo grouping functions are presented, and the residuated implication (co-implication) operators derived from them are investigated. Not only that, some applications of pseudo overlap (grouping) functions in multi-attribute (group) decision-making, fuzzy mathematical morphology and image processing are discussed. Experimental results show that, in many application ﬁelds, pseudo overlap functions and pseudo grouping functions have greater ﬂexibility and practicability.


Introduction
The aggregation function (or aggregation operator) is an important concept in decision theory, information fusion and fuzzy inference systems etc. (see [1,2]). The process of combining several (numerical) values into a single representative one is called aggregation, and the function performing this process is called aggregation function. As a special aggregation operator, the concept of overlap functions (OF for short) is proposed in [3] by scholars such as H. Bustince. Since then, OFs as well as associated grouping functions have become a new hot direction, with rich research achievements (see [4][5][6][7][8][9][10]), and have been applied in many fields, such as multi-attribute (group) decision-making, rule-based classification and image processing etc. (see [11][12][13]).
In recent years, in order to expand the application scope of OFs and grouping functions, many scholars have put forward some more general concepts, such as r-(quasi-)overlap functions [14] etc. However, all of them keep the requirement of commutativity (symmetry). This paper mainly removes the commutativity of OFs as well as grouping functions, thus studies more general aggregation functions, that is, pseudo overlap functions as well as pseudo grouping functions.
It is necessary to emphasize the motivation for studying pseudo overlap (grouping) functions. Our main starting points include the following aspects: (1) For aggregation functions, the commutativity is not always necessary, for example, a copula is a kind of aggregation function which is closely related to overlap function, but a copula could be non-commutative (see [15]). So, it is natural to investigate noncommutative overlap (grouping) functions; (2) From the perspective of fuzzy logic theory, overlap functions are closely related to t-norms and fuzzy reasoning. The notion of t-norm is generalized to the non-commutative case, that is, pseudo t-norm, which plays an important role in non-commutative fuzzy logic (see [16][17][18]). Therefore, and in the same way, it becomes natural to study pseudo overlap (grouping) functions; (3) The overlap functions can be applied to multi-attribute (group) decision problems, but when multiple attributes (or multiple decision makers) have different importance, overlap functions may display some limitations due to the commutativity. Then, it is necessary to extend overlap function to non-commutative case. In fact, I. A. Da Silva et al. constructed a kind of weighted average operators based on n-dimensional overlap functions (see Theorem 3.1 in [13]), which are non-commutative overlap functions when a weak condition is attached; (4) In the theory of mathematical morphology, t-norms and fuzzy implication operators are the basic tools (see [19][20][21][22][23]). Similar to fuzzy morphology based on t-norms, we can construct a new mathematical morphology based on overlap functions. More generally, since the theory of mathematical morphology endowing non-commutative operators has wider applicability (see [24,25]), the establishment of corresponding fuzzy erosion and dilation operators considering pseudo overlap (grouping) functions can expand the existing fuzzy mathematical morphology theory; (5) Moreover, grouping functions can be applied to image segmentation [11,12], but for image processing, the target object and background are usually unequal. In order to express such inequality (or asymmetry), we naturally select non-commutative operators, that is, pseudo grouping functions. We have reason to believe that we can obtain a better effect when we use pseudo grouping functions to edge extraction and threshold segmentation of images.
Based on the above considerations, we will study the basic properties and some applications of pseudo overlap (grouping) functions in detail. This paper is organized as follows. In Section 2, we review some preliminary concepts (pseudo t-norm, pseudo t-conorm, copula, overlap function, grouping function, fuzzy implication and residuated lattice). For the third part, we give the notion of (n-dimension) pseudo overlap functions as well as some concrete examples, provide some construction methods, illustrate the relevant conclusions of residuated implications induced by pseudo overlap functions, and analyze the relationship between overlap functions and residuated lattices. In Section 4, we investigate pseudo grouping functions and co-implications derived from them. Section 5 presents some applications of pseudo overlap (grouping) functions in multi-attribute group decision making (MAGDM for short), fuzzy mathematical morphology and image processing. Finally, conclusions are declared in Section 6.

Preliminaries
In this part we state a few basic notions as well as results.

Pseudo Overlap Functions and Their Residuated Implications
In this section, we consider existing outcomes of residuated implication to introduce the concept of pseudo overlap functions and residuated implication derived from them. Then, we elaborate the relationship between pseudo overlap functions and continuous pseudo t-norms and copulas, and expand the dimension to explain the definition of multidimensional pseudo overlap functions. After that, we also illustrate the general construction method of pseudo overlap functions, and finally provide some examples to explain in detail.
(3) if O is pseudo t-norm, then it is a copula if and only if it satisfies Lipschitz property with constant 1 (that is, ∀x 1 , x 2 , y 1 , Proof. (1) It follows from Definitions 7 and 11; (2) If O is pseudo t-norm, since O is a pseudo overlap function, by Definition 11 (O4'), O is continuous. Using Lemma 1, we know that O is commutative. By (1), O meets (O1) ∼ (O5). Thus, O is a t-norm satisfying continuity and has no nontrivial zero factors; (3) Suppose that O is a pseudo overlap function and pseudo t-norm. By (2) we know that O is a positive continuous t-norms. Applying Theorem 1 in [15], we can get that O is a copula if and only if it satisfies Lipschitz property with constant 1.
According to the proof of Theorem 2 in [15], we get that every positive copula is a pseudo overlap function. Therefore, we obtain the diagram in Figure 2 with the relationship between some concepts. The following example shows that there are some POFs and pseudo t-norms which are not copulas.
is an n-dimension pseudo overlap function.
is an n-dimension pseudo overlap function.
Next, we provide some construction methods and theorems of POFs.

Proof. Evidently, O is continuous and monotonous. Due to
is a pseudo overlap function.

Proof. One easily verifies that O is continuous and monotonous. Then
In the existing literature, some scholars have shown that fuzzy conjunctions can induce residual implication, such as t-norms, t-conorms and overlap functions etc. Additionally, some conjunctions can also induce two residuated implications by removing commutativity, such as pseudo t-norms and pseudo t-conorms. Moreover, since the function can still induce fuzzy implication without commutativity, we can define two residuated implications which satisfy the residual property induced by pseudo overlap function, namely residuated Obviously, R O if the considered pseudo overlap function is commutative. Binary functions R (1) O is infinitely ∨-distributive in its first variable; (2) O(x, y) ≤ z when and only when x ≤ R Moreover, conditions as below are equivalent: O (x, y)) ≤ y when taking any x, y ∈ [0, 1]; Proof. By Theorem 4.1 in [34] (or Theorem 3.1 in [35]), we get that the conditions (1) ∼ (4) are equivalent. By Theorem 4.2 in [34], we get that the conditions (1') ∼ (4') are equivalent.
In the following table, we provide some examples of pseudo overlap functions and their residuated implications (see Table 1).
In Table 1 . In the following, some properties that pseudo overlap functions and residuated impli- O satisfy are presented.
Then statements as below hold: O satisfies (CB) if O has unit element 1. Table 1. Examples of pseudo overlap functions and their residuated implications.

Remark 1. Since the two functions R
(1) O and R (2) O are induced by the pseudo overlap function without commutativity, the above proposition can also expand some properties, such as:

Pseudo Grouping Functions and Their Residuated Co-Implications
In this section, we show notions of pseudo grouping functions as well as discuss residuated co-implications (the notion of co-implication was first proposed by B. De Baets and J. Fodor in [29]; it is also called deresiduum, see [35]). In addition, we also provide the general construction method of pseudo grouping functions. Finally, we provide some detailed examples. Observe that a PGF can be obtained by duality from a pseudo overlap function. We have the following basic result.
Then we will provide some examples of n-dimensional PGFs.
is an n-dimensional PGF.
Obviously, R G when the pseudo grouping function G is commutative. The functions R (1) G and R (2) G are called residuated co-implications associated with the pseudo grouping function G on the first and second variables, respectively. Theorem 9. Let G be a PGF on [0, 1], statements as follows are equivalent: (1) G is infinitely ∧-distributive in its first variable; (2) G(x, y) ≥ z when and only when x ≥ R (1 ) G is infinitely ∧-distributive in its second variable; (2 ) G(x, y) ≥ z iff y ≥ R (2) G (x, z) when taking arbitrary x, y, z ∈ [0, 1]; G (x, y)) ≥ y for arbitrary x, y ∈ [0, 1]; (4 ) R The concrete examples of pseudo grouping functions and residuated co-implications corresponding to them are as follows (see Table 2). Table 2. Examples of pseudo grouping functions and their residuated co-implications.

Pseudo Grouping Function
Residuated Co-Implications

Some Applications of Pseudo Overlap Functions
In this section, we reveal some applications of POFs (PGFs) in MAGDM, fuzzy morphology and image processing.

Pseudo Overlap Functions Generated by n-Dimensional Overlap Functions
Overlap functions have been used in multiple-attribute decision making, but because they are commutative, they cannot be directly applied to the decision problems in which the importance degree of different attributes is different. At the same time, in group decisionmaking, if the importance of different decision experts is different, the overlap function cannot be directly used to aggregate their decision information.

MAGDM on Account of Pseudo Overlap Functions
Similar to the method in [13], we present a solution for the problem as below: (1) Standardize the decision matrix S (k) to a normal decision matrix N (k) = (n ij , j / ∈ I ; (2) Given a t-dimensional POF O, the standard decision matrices are aggregated into the overall decision matrix with O as follows: For every scheme x i , in order to calculate the group totality preference value gtp i , we aggregate its membership degrees to each attribute by using an m-dimensional pseudo overlap function O as follows: gtp i = O (c i1 , · · · , c im ); (4) Array all schemes in descending order according to group totality preference values and select the top scheme.

Living example
The living example in [13] is regarded to demonstrate our approach. Assume an investor plans to invest a portion of capital in one corporation. Finally, he narrows the range to six corporations: x 1 stands for the chemical corporation, x 2 stands for the food corporation, x 3 stands for the computer corporation, x 4 stands for the car corporation, x 5 stands for the furniture corporation, x 6 stands for the pharmaceutical corporation. The investor is assisted by three experts (e 1 , e 2 and e 3 ). The experts establish some attributes utilized to estimate investments.
The decision matrices of the three experts are as follows, see Tables 3-5.   These decision matrices are normalized respectively to obtain three normal decision matrices N (1) , N (2) and N (3) , which are described as below, see Tables 6-8.  Next, because three experts with the weights w = (0.3, 0.3, 0.4), we aggregate the standard decision matrices of them by using three-dimensional pseudo overlap function O 1 in Example 6 to obtain the overall decision matrix, as shown in Table 9 below. Then we obtain the group totality preference vector GTP by considering O 6 in Example 7, in which w = (0.1, 0.1, 0.2, 0.2, 0.2, 0.2). The outcome is shown in Table 10. On account of Table 10, we get the array of the schemes as below: Of course, we can get other rankings by using other different pseudo overlap functions in Example 6 and 7. Table 11 contains eleven rankings obtained by the pseudo overlap functions and the result of four methods in [13] for this matter. Table 11. Summary of the permutations gained with presented means as well as other methods.

Six-Dimensional Functions Ranking
Maximum The above table shows that fifteen methods return ten different rankings. In addition, when there are no exact weights, other non-commutative pseudo overlap functions (which cannot be directly generated by the method in Theorem 10) can also be selected. Non-commutative functions represent the different importance of different attributes or experts, but the importance degrees are not expressed by specific numbers, which are hidden in the functions (in this case, we can think that the weights can vary according to the attribute value). For the example above, we can use the following noncommutative pseudo overlap functions: The results are shown in Table 12 below. Analyzing Table 12, it is obvious that the nine schemes revert to four various arrangements. Among them, most (seven) think the best alternative is x 2 and nine think that x 3 is the worst alternative. Moreover, rankings in Tables 11 and 12 can be fused and further analyzed by referring to the method in [36], which is omitted here.
Through the above comparative analysis, we show that the pseudo overlap functions can not only aggregate multiple information, but also indicate the importance of different information (including the importance of attributes and the importance of experts). They are more flexible than the overlap functions, and convenient for decision makers for choosing good alternatives.

Application of Pseudo Overlap Functions in Fuzzy Mathematical Morphology
The method of analyzing object shape in images using different neighborhood transformations based on set theory is known as mathematical morphology. At present, there is a mathematical morphology for binary images and a fuzzy mathematical morphology for gray images. Some authors improve fuzzy mathematical morphology by quoting t-norms and fuzzy implications when defining the most basic fuzzy dilation and fuzzy erosion operations (see [37]). We will introduce pseudo overlap functions and pseudo grouping functions into fuzzy mathematical morphology, and use the new operators in fuzzy mathematical morphology to extract the edge of gray images. When a pseudo overlap function is applied to fuzzy mathematical morphology, the pseudo overlap function is used to replace the intersection operation in fuzzy dilation, and its corresponding pseudo grouping function is used to replace the union operation in fuzzy erosion. At this time, the gray image A is subjected to the fuzzy expansion operation D(A, B) and the fuzzy corrosion operation E(A, B) performed by the structural element B, which are defined as follows, respectively:

Experimental Framework
In the original fuzzy mathematical morphology, after the image is blurred, t-norms and fuzzy implications are often used to expand and erode the image, and then de-blur it. However, we use pseudo overlap functions and pseudo grouping functions as fuzzy expansion and erosion operators, convex combine their results, and then defuzzify it. The specific steps are as follows.
(1) Fuzzy grayscale image The grayscale image shown in Figure 3 is selected as the image to be processed, which is a single channel image with a resolution of 1200 * 675. (3) Defuzzification convex combination result Firstly, the results of dilation and erosion are obtained through the convex combination formula shown below to obtain the fuzzy set C.
Then, the fuzzy set C is defuzzified to obtain the number set M .
where is rounding down. After the fuzzy expansion and fuzzy erosion results of the original intensity of the image are convex combined, the convex combination results are de-blurred to obtain M , as the grayscale image shown in Figure 4.

Analysis of Experimental Results
In order to present the actual application effect, we used the edge extraction operation in mathematical morphology to extract the edges of objects in the original gray image M and the gray image M processed by fuzzy mathematical morphology based on pseudo overlap functions. The principle of edge extraction of a binary image by mathematical morphology is to dilate the foreground of the binary image, and then make a difference between the dilation result and the original image to obtain the edge of the binary image.
Firstly, binarization is performed on the gray image M and M , respectively, to obtain the binary images M b , M b , as shown in Figures 7 and 8 below.  We take the binary structure operator B b as follows: Then we perform edge extraction operations on binary images M b and M b , respectively: The results of edge extraction for the original gray image and the gray image processed by fuzzy mathematical morphology with pseudo overlap functions are revealed in Figure 9 and Figure 10 respectively.  According to Figures 9 and 10, the edge graph obtained by our method (Figure 10) has fewer unorganized points, that is, the gray image processed by fuzzy mathematical morphology with pseudo overlap functions can achieve better image edges through edge extraction in mathematical morphology. This is because the dilation and erosion operations of fuzzy mathematical morphology with pseudo overlap function expand the contrast of foreground and background in the gray image from two aspects: reducing the background gray value and increasing the foreground gray value, which can be seen from Figures 5 and 6. Moreover, after introducing convex combination, the advantages of dilation and erosion are combined to further expand the contrast of foreground and back-ground in the gray image. This allows us to extract the foreground with a more accurate threshold in the binarization operation, thus reducing noise similar to that in Figure 9.

Application of Pseudo Grouping Functions in Image Processing
In image segmentation, one of the most commonly used methods is threshold segmentation. In threshold processing, different targets of a picture are described by the gray level of every pixel. This technology mainly looks for a threshold t, so that the pixels with an intensity less than or equal to t are classified as the backdrop of the picture, while pixels with an intensity greater than t are classified as targets, and the converse is also true ( [38]). Aranzazu Jurio et al. proposed a threshold algorithm based on the convex combination of the fuzzy method and grouping function in reference [12]. Our work is to introduce the pseudo grouping function to calculate the grouping value. Finally, we use the improved threshold algorithm to perform threshold segmentation on a gray image and compare it with Otus's algorithm, one of the most commonly used thresholding methods.
In this section, we show the performance of thresholding segmentation with the convex combination of pseudo grouping functions and the convex combination of grouping functions on 10 T1 weighted MRI images (see Figure 11). The MR cerebrum information as well as hand-operated segmentations originated from the morphometric analysis center of Massachusetts General Hospital. We evaluated the quality of excision by comparing with the ideal manually segmented images obtained on the same web page (see Figure 12). The purpose of this image segmentation was to divide each pixel in the cerebrum into two kinds-gray matter and white matter. In fact, it is a part of brain region volume analysis, which plays a great role in assessing the development of illness, for instance, Alzheimer's disease, epilepsy or schizophrenia ( [39,40]).  To improve the effect of thenconvex combination thresholding results of grouping functions on image segmentation, we consider using pseudo grouping functions instead of grouping functions. Because theoretically a grouping function is a special case of a pseudo grouping function, pseudo grouping functions give a wider function selection range and have a greater chance of finding a better function, which may improve the accuracy of thresholding image segmentation. We confirm this below.
The specific content of the experiment is that we randomly selected the grouping functions in advance and segmented the 10 images with their convex combination, which achieved a certain segmentation effect. Then, the convex combination of pseudo grouping functions was also used for segmentation to compare with it. Finally, we illustrate the superiority of the proposed method by the percentage of good classified pixels.
In the thresholding process of Aranzazu jurio et al., a fuzzy set µ Q B t (q) related to the background and a fuzzy set µ Q O t (q) related to the image target are constructed by using the strict equivalence function for a fixed gray level t(t = 0, 1, . . . , L − 1), for each gray level q(q = 0, 1, . . . , L − 1), h(q) is expressed as quantity of pixels with pixel intensity q.
When calculating the maximum grouping value using the convex combination of grouping functions G G comb (x, y), select an appropriate threshold for each image: when the convex combination of pseudo grouping functions PG G comb (x, y) is introduced for the thresholding operation, the threshold t * related to the maximum grouping value is taken as the best threshold. The formula of the improved algorithm is: Then, five grouping functions are used as follows: (1−x)(1−y) . The five pseudo grouping functions are selected as below: PG 1 (x, y) = G 1 (x, y) PG 2 (x, y) = G 2 (x, y) PG 3 (x, y) = G 3 (x, y) otherwise PG 5 (x, y) = G 5 (x, y). Through the convex combination of the above grouping functions and pseudo grouping functions, the results of threshold segmentations of 10 graphs are as follows (see Figures 13 and 14).  We show the experimental results through specific data. In Table 13, we reveal the threshold obtained from every picture as well as the percentage of good sorted picture elements. The second column presents the outcomes gained from the convex association of grouping functions, and the third column presents the outcomes gained from the convex association of PGFs. As we can see from the experiment, our method has higher accuracy, that is, the use of convex association of PGFs significantly improves the segmentation accuracy of each image. Then, in contrast with Otsu's algorithm ( [41]), the thresholding method on the basis of pseudo grouping functions has greater advantages. In Table 14, we show the thresholds obtained by the two algorithms and the percentage of pixels with good classification. We can find that our method obtains the better results for each picture in the test case, hence it mends Otsu's thresholding technique with regard to the group of pictures.

Conclusions
Starting from many practical application fields, we expand the theory of OFs as well as grouping functions, introduce new concepts of POFs and PGFs for the first time, and investigate their basic properties, construction methods, and induced residuated implication operators and residuated co-implication operators. Moreover, we discuss some applications of pseudo overlap (grouping) functions in multiple attributes group decision-making, fuzzy mathematical morphology and image processing. The experimental results show that: (1) pseudo overlap functions can aggregate the decision information of attributes or experts with different importance; (2) pseudo overlap functions and their convex combination can be used to enrich the operators in fuzzy mathematical morphology, and expand the contrast between foreground and background in gray images, so as to reduce the noise in edge extraction; (3) pseudo grouping functions can be used to improve the accuracy of thresholding image segmentation in image processing. Therefore, POFs and PGFs have a wide adaptability and application value. There are still many theoretical and applied problems to be studied in terms of pseudo overlap functions and fuzzy logic (see [42][43][44][45][46][47]).