Time-Leveled Hypersoft Matrix, Level Cuts, Operators, and COVID-19 Collective Patient Health State Ranking Model

Tis article is the frst step to formulate such higher dimensional mathematical structures in the extended fuzzy set theory that includes time as a fundamental source of variation. To deal with such higher dimensional information, some modern data processing structures had to be built. Classical matrices (connecting equations and variables through rows and columns) are a limited approach to organizing higher dimensional data, composed of scattered information in numerous forms and vague appearances that difer on time levels. To extend the approach of organizing and classifying the higher dimensional information in terms of specifc time levels, this unique plithogenic crisp time-leveled hypersoft-matrix (PCTLHS matrix) model is introduced. Tis hypersoft matrix has multiple parallel layers that describe parallel universes/realities/information on some specifc time levels as a combined view of events. Furthermore, a specifc kind of view of the matrix is described as a top view. According to this view, i -level cuts, sublevel cuts, and sub-sublevel cuts are introduced. Tese level cuts sort the clusters of information initially, subjectwise then attribute-wise, and fnally time-wise. Tese level cuts are such matrix layers that focus on one required piece of information while allowing the variation of others, which is like viewing higher dimensional images in lower dimensions as a single layer of the PCTLHS matrix. In addition, some local aggregation operators are designed to unify i-level cuts. Tese local operators serve the purpose of unifying the material bodies of the universe. Tis means that all elements of the universe are fused and represented as a single body of matter, refecting multiple attributes on diferent time planes. Tis is how the concept of a unifed global matter (something like dark matter) is visualized. Finally, to describe the model in detail, a numerical example is constructed to organize and classify the states of patients with COVID-19.


Introduction
Te discipline of modeling and decision-making in an uncertain and ambiguous environment is an incredible endeavor for the human mind. To optimize the feld of modeling and decision-making in an uncertain and ambiguous universe, Zadeh [1] developed the fuzzy set. Te author extended the crisp state (either yes or no) of the human mind by introducing fuzziness into mathematical structures, i.e., partial yes and partial no represented by some membership and nonmembership in the area of decisionmaking. Later, in 1986, Atanassov [2] extended this state of vagueness further by introducing intuition, or hesitation, into decision-making structures, which were called intuitionistic fuzzy set theory. Tis means adding some layers of hesitation with the layers of partial yes and partial no. Tese three states of the human mind were represented by the degree of membership, the degree of nonmembership, and the degree of hesitation. In addition, Atanassov [3] introduced an interval-valued fuzzy set (IVFS) in 1999, which is somewhat extended form of IFS (hesitating partial yes and partial no values packed in interval units). However, these extended set theories were unable to deal with uncertainty. To deal with uncertainty, Smarandache [4] introduced neutrosophy by generalizing hesitation as an independent indeterminate neutral factor and mentioning that the neutrosophic set is a generalization of the intuitionistic fuzzy set, the inconsistent intuitionistic fuzzy set, and the Pythagorean fuzzy set, and some other applications of the neutrosophic set were discussed by Smarandache and coauthors [5,6]. Later, Molodtsov formulated a soft set in 1999 [7][8][9][10], in which the author further extended this extended fuzzy theory by considering multiple attributes parameterized by multiple subjects. Later, in 2018 [11], Smarandache further expanded this soft set to a hypersoft set and a plithogenic hypersoft set by splitting the attributes into various levels of attributes called subattributes. Te author presented the hypersoft set as a set containing many subjects (matter bodies) parametrized by a combination of attributes/subattributes. In the case of the hypersoft set, the observer experiences an outside perception of the information that can be displayed in any extended fuzzy environment such as fuzzy, intuitionistic, and neutrosophic. Te plithogenic hypersoft set is a set whose elements are structured by one or more attributes, and each attribute can have many values such that each attribute value has a corresponding degree of membership of an element x (subject) to the set P, with respect to each separately given criterion. It is an extended version of the hypersoft set and more widely applicable since the observer can intrinsically see the state of the element x (subject) by looking at each attribute separately. Smarandache introduced plithogeny [12][13][14] and expanded the vision of an ambiguous, uncertain universe. Te author raised some open problems in this extended fuzzy set theory such as designing multicriterion decision-making techniques and construction of operators. Rana et al. [15] addressed these open problems and expanded the dimension of the plithogenic hypersoft set [11] by introducing and representing these PFWHSS in a matrix form called the plithogenic fuzzy whole hypersoft matrix. In addition, some local aggregation operators have been developed for the plithogenic fuzzy hypersoft set (PFHSS). Tis matrix was developed for a specifc combination of attributes and subattributes, which was a limited structure designed for a single combination of attributes and subattributes, and a general model was required to be formulated. Terefore, later, Rana et al. [16] generalized the plithogenic whole hypersoft matrix to an extended form of the matrix called the plithogenic subjective hyper-supersoft matrix. It is a generalized form and a matrix superior to the previously developed matrix. It has a greater capacity to express the variations of connected attributive levels as well. Tese attributive levels are represented in the form of matrix layers. Te application of this matrix is provided in the form of a new ranking model called the plithogenic subjective local-global universal ranking model. Some other researchers also addressed this extended fuzzy set theory. Saeed et al. [17] discussed the prognosis of allergy-based diseases by using Pythagorean fuzzy hypersoft and recommended medication. Muhammad et al. and Zulqarna et al. [18][19][20][21] described a correlation coefcientbased decision-making approach and discussed some of its properties for interval-valued neutrosophic hypersoft sets. Te authors also outlined some basic operations and discussed their properties for interval-valued neutrosophic hypersoft sets. Furthermore, generalized aggregation operators have been established for neutrosophic hypersoft sets, and robust aggregation operators have been established for intuitionistic fuzzy hypersoft sets. Yolcu and Ozturk [22] described fuzzy hypersoft sets and discussed their application as a decision-making model. Siddique et al. [23] established a multicriterion decision-making approach for aggregation operators by using Pythagorean fuzzy hypersoft sets. Abdel-Basset et al. [24] designed an integrated plithogenic MCDM approach for the fnancial performance evaluation of manufacturing industries. Grida et al. [25] evaluated the impact of COVID-19 prevention policies on aspects of the supply chain under uncertainty. Akram et al. stated that after all these developments, there is still a vacuum in this extended fuzzy set theory. As we know, the source of the variation of the universe as information, data, and realities is time. It is a grim need to discuss and include this source of variation in the extended fuzzy set theory, and it is essential to design such mathematical higher dimensional structures that include time as a variation unit.
Tis recent paper is a frst step toward designing such higher dimensional mathematical structures that include time as a fundamental source of variation. It further provides an upgraded and broadened plithogenic universal model by introducing a time-leveled variation in a hypersoft matrix, which deals with data and information as a magnifed angle of vision. As we know, most of the variations in this universe are time-dependent like weather graphs, stock exchange, and website ratings, therefore, it is of tremendous help if we use a plithogenic hypersoft matrix to cope with scattered time-varying pieces of information of the plithogenic universe in crisp and other environments (i.e., fuzzy and neutrosophic environments). Initially, a three-dimensional broadened view of the PCHS-Matrix is portrayed to represent the plithogenic crisp time-leveled hypersoft set. Tis PCTLHS matrix is a rank-three tensor that shows three types of variations. It contains several matrix layers, whereas each layer is a tensor of rank two (i.e., a case of the ordinary matrix) expressed in the crisp environment. Furthermore, this PCTLHS matrix represents time-dependent multiple parallel universes/parallel realities/information. For example, if we are organizing the information of COVID-19 patients (subjects) admitted to a certain hospital, the record of their symptoms (attributes) at some time levels can be organized as a PCTLHS matrix. By using this connected matrix expression, we can see and classify all information immediately; i.e., the information from a group of patients is assigned to a combination of attributes and observed at diferent time levels. Te three types of PCTLHS matrix variation indices would describe patients (subjects), their symptoms (attributes), and the time-leveled states of their symptoms as time-based subattributes, whereas one kind of level cuts (i-level cuts) related to the top view of the PCTLHS matrix focuses on the subject (patient) separately and would display their attributes (symptoms) at several time levels as matrix layers. In this way, these level cuts can focus on required information while displaying the variation of other information as a single matrix layer of the hypersoft matrix. Furthermore, these level cuts further cleave into sublevel cuts by splitting the matrix layer at one of the two remaining variation indices, whereas these sublevel cuts do ofer the display of the previous lower dimension in the further lower dimension and enable us to sneak in an inside view of the expanded universe; i.e., after focusing on a subject explicitly through an i-level cut (single layer of the layered matrix), our next focus is on the attribute (a specifc symptom) of that subject (patient) through the sublevel cut (row or a column of the one layer of the multiple layered matrix). One may call the expanded view of the matrix an implicit view of the universe. In the next stage of these sub-level cuts, sub-sublevel cuts are constructed by splitting the sublevel cuts (row or column of a specifc layer of the PCTLHS matrix) at the third variation index of the matrix, which means after focusing on the subject (patient) and attribute (symptom) through the level cut and sublevel cut, our next focus is on the specifc time level of information. After applying all splits to all indices, outcomes would be refected as singletons. Tese level cuts, sublevel cuts, and sub-sublevel cuts are displaying their traits as zoom-in and zoom-out functions to provide the interior and exterior view of these timebounded events, which can be argued as a contraction of the expanded higher dimensional universe. Te sub-sublevel cuts provide a contract picture of the smallest part of a single or multiple universe. In this way, the expanded universe of matrix layers could be contracted to a single point. Similarly, by reversing the process, one can expand the same singleton into higher dimensions of rows, columns, matrices, matrix layers, and clusters of matrix layers. Te matrix expression is the more appropriate expression to represent multidimensional data compared to the classic set expression. Te question now arises as to why a collective state ranking would be preferable to an individual state ranking. Te answer is precise and obvious since the collective state ranking sorts attributes of a group of people (subjects) as compared to an individual state ranking in which attributes of a person (subject) are categorized. By using the collective state ranking, one can categorize attributes on a broader spectrum such as attributes (health states) of a group of patients that can be distinguished, and hence, remedy for the most dominant attribute can be identifed. Ten, the cure for that attribute, which would be cough syrup, would be introduced to the market and manufactured on a large scale. Terefore, by considering a collective state ranking, any product associated with the dominant attribute may be imported or manufactured on a larger scale, which is not possible when considering an individual state ranking.
In the fnal stage, plithogenic local aggregation operators are developed and utilized to elaborate the activity of these several types of level cuts based on variation indices. Tese local operators serve the purpose of the unifcation of matter bodies in the universe, which means all their attributes are observed as they are refected from a single entity. In this way, attributes are focused and subjects (matter bodies) are merged. Tis means that all elements of the universe are fused and represented as a single body of matter refecting multiple attributes at diferent time levels. Tese i-level cuts unify the subjects using the aggregation operators, which help introduce the concept of a unifed global matter that appears like dark matter that can have attributes but not individual bodies of matter. Tese operators are named plithogenic disjunction, plithogenic conjunction, and plithogenic averaging operators. For further precise applications of the model, a numerical example is derived for the organization and classifcation of COVID-19 data or information at two distinct time levels.
Tis article is organized into seven basic sections. After the introduction (Section 1), Section 2 summarizes some related preliminaries. Section 3 introduces some fundamental new concepts and defnitions of the PTCHS-Matrix. Section 4 gives a mathematical description of the top-tobottom view of the PLCHS-Matrix, and its splits major structures (connected matrix layers) into smaller structures such as i-level cuts, sublevel cuts, and sub-sublevel cuts. Section 5 describes the application of these split structures as the health state model for patients withCOVID-19 by using PCTLHS-Matrix. Section 6 introduces the set-based operations for the unifcation of i-level cuts, sublevel cuts, and sub-sublevel cuts of the PCTLHS matrix. Section 7 summarizes some conclusions and discussions on future research.

Preliminaries
Tis section summarizes some basic defnitions of soft sets, hypersoft sets, crisp hypersoft sets, plithogenic hypersoft sets, and plithogenic crisp hypersoft Sets. Tese defnitions would help expand the theory of plithogenesis.
Defnition 1 (see [7]). (Soft set) Let U be the initial Universe of discourse and E be a set of parameters or attributes with respect to U, let P(U) denote the power set of U, and A ⊆ E is a set of attributes. Ten, the pair (F, A) where F: A ⟶ P(U) is called soft set over U. In other words, a soft set (F, A) over U is a parameterized family of subsets of U. For e ∈ A, F(e) may be considered as a set of e elements or e approximate elements: Defnition 2 (see [11]). (Hypersoft set)Let U be the initial universe of discourse and P(U) be the power set of U.
Let a 1 , a 2 , . . . , a n for n ≥ 1 be n distinct attributes, whose corresponding attributes values are, respectively, the sets Applied Computational Intelligence and Soft Computing is called a hypersoft set overU.
Defnition 3 (see [11]). (Plithogenic crisp hypersoft set) Let U c be the initial crisp universe of discourse and P(U c ) be the power set of U. Let a 1 , a 2 , . . . , a n for n ≥ 1 be n distinct attributes, whose corresponding attributes values are, respectively, the sets Defnition 4 (see [25,26]). (Supermatrices) Square or rectangular arrangements of numbers in rows and columns are matrices, and we call them simple matrices, while the supermatrix is the one whose elements are themselves matrices with elements that can be either scalars or other matrices.
a � a 11 a 12 where where a is a supermatrix. Note: the elements of supermatrices are called submatrices; i.e., a 11 , a 12 , a 21 , a 22 are submatrices of the supermatrix a. In this example, the order of the supermatrix a is 2 × 2, and the order of submatrices a 11 is 2 × 2,a 12 is 2 × 2, anda 21 is 3 × 2, and the order of the submatrix a 22 is 3 × 2, and we can see that the order of the supermatrix does not tell us about the order of its submatrices.

(PCTLHS Matrix).
Let U C (X) be the crisp universe of discourse and P(U C ) be the power set ofU C . A k j is a combination of attributes and subattributes (time-leveled attributes) for some j � 1, 2, 3, . . . , N represents N number of attributes, k � 1, 2, 3, . . . , L represents L number of time levels and x i i � 1, 2, 3, . . . , and M represents the M number of subjects under consideration. Ten, plithogenic crisp time-leveled hypersoft matrix (PCTLHS matrix) is a mapping from the cross product of attributes/time-leveled attributes on the power set of the universe P(U C ) , represented in the matrix form. Tis mapping with its matrix form is described in equations as follows: where In simple words, a plithogenic crisp hypersoft set, represented in the matrix form, is called plithogenic crisp time-lined hypersoft matrix (PCTLHS matrix).
Tis matrix has three possible expansions associated with its three-dimensional views, which are described in crisp environments.

Tree-Dimensional Views and Level Cuts of the PCTLHS Matrix.
As we know, all ordinary matrices in the real vector space are rank 2 tensors. Similarly, as an extended matrix version, the PCTLHS matrix with its three variation indices is a rank 3 tensor. Te PCTLHS matrix contains layers of ordinary matrices called matrix layers or level cuts.
For a detailed description, we consider the example of the PCTLHS matrix A � [A ijk ]. Te index i refers to variations of rows used to represent the subjects under consideration, j specifes a variation of columns used to represent attributes of subjects, and k provides variations of layers of rows and columns that would be used to represent the attributes on specifc time levels (varying matrix layers as clusters of rows and columns). Similarly, [A jki ] is interpreted as the index, and j ofers a variation of rows, k gives a variation of columns, and i ofers variation of clusters of rows and columns.
Te PCTLHS matrix contains layers of ordinary matrices, termed matrix layers or level cuts of the PCTLHS matrix Te level cuts, sublevel cuts, and sub-sublevel cuts would be defned by specifying variation indices i, j, k for their positive integer values.

Level Cuts.
Level cuts are submatrices (frst-level splits) of the PCTLHS matrix that can be further described as parallel matrix layers. Te PCTLHS matrix is generated by uniting these matrix layers. Tese level cuts of the PCTLHS matrix are obtained by assigning a specifc integer value to the frst variation index at a time. According to three types of view of the PCTLHS matrix, level cuts are distributed in three categories, i.e., i-level cuts j-level cuts k-level cuts Note: in this article, only the frst type of level cuts are formulated and explored, whereas the other two types would be discussed in the upcoming version.

Sublevel
Cuts. Sublevel cuts are perceived as level Cuts of level cuts (second splits applied over frst splits) of the PCTLHS matrix. Tese sublevel cuts are columns or rows of the submatrix ( the matrix obtained after the frst split). Te sublevel cuts are obtained by assigning a specifc integer value to one of the two variation indices of a parallel layer (submatrix) of the PCHS matrix.

Sub-Sublevel Cuts.
Sub-sublevel cuts are obtained by assigning a specifc integer value to the variation index of sublevel cuts (the third-level split over the second split). Te sub-sublevel cut is one specifc element (point) of the sublevel cut (column or row). Tese level cuts, sublevel cuts, and sub-sublevel cuts are images of the higher dimensional universe in lower dimensions and can be used as tools for getting images and transformations. Te detailed classifcation of these level cuts, sublevel cuts, and sub-sublevel cuts is described below.
Te utilization of these cuts is that one can contract the expanded dimension of the PCTLHS matrix to a matrix, then to a row or column matrix, and then further to a single point. Similarly, the reverse procedure would provide an expansion of the universe.

Variation Indices of the PCTLHS Matrix.
Tree types of variation indices are used to represent a general element (μ A k j (x i )) of the PCHS matrix. Te frst variation index i (associated with subjects) represents M rows of an M × N submatrix which is a single layer of the M × N × L PCHS matrix. Te second variation j (used to specify attributes) represents N columns of this submatrix. A third variation index k (represent attributive levels) represents L layers or L level cuts of the M × N × L PCHS matrix. Te index-based views, level cuts, sublevel cuts, and Sub-sublevel cuts are categorized into the following types.

i-Level Cuts, Sublevel Cuts, and Sub-Sublevel Cuts of the PCTLHS Matrix
Te top-to-bottom view of the PCTLHS matrix consists of top to the bottom layer of the matrix. Tese layers are formulated by specifying the variation index i. Equation (8a) describes the top-to-bottom view of the PCTLHS matrix: Te top-down view of the PCTLHS matrix in an expanded form is described as follows:  (9) represent ilevel cuts. Furthermore, these i-level cuts focus on each subject (patient) separately and display variations of their attributes (symptoms) at several time levels.

i-Level Cuts
A
Te i-level cuts of the PCTLHS matrix, constructed by splitting the index i, are called i-leveled splits or i-level cuts.
i-level cuts A [i] of the PCTLHS matrix (equations (8a) and (8b)) for fxed i � 1, 2, . . . , M are given as follows: Tese i-level slices are submatrices of the PCTLHS matrix, which can be further described as parallel layers of the matrix. Te PCTLHS matrix is created by uniting these matrix layers. It is observed that i-level cuts can focus and classify subjects (matter bodies) in scattered clusters of information. For example, the frst i-level cut A [1] focuses on the frst subject x 1 and displays the N-number of attributes at L-number of time levels. Similarly, the second i-level cut A [2] focuses on the second subject x 2 and A [M] focuses on Mth subject x M .

i j -Sublevel
Cuts. i j -sublevel cuts are constructed by frst specifying i � m where m is any positive integer from 1 to M, then further specifying j � n such that n is a positive integer from 1 to N, and then varying k � 1, 2, . . . , L as given in equation (12). Te observer can sort the clusters of information frst subject-wise and then attribute-wise, respectively, by using this top-down view of the PCTLHS matrix Tese i j -sublevel cuts are the column-wise split of each topto-bottom layer of the matrix: It is observed that after classifying information subjectwise through i-level cuts, one would further distinguish information attribute-wise by using i j -sublevel cuts. For example, for a fxed i � 1 and j � 1, a sublevel cut is a column matrix, displaying the frst subject's information for its frst attribute at k time levels where k � 1, 2, . . ., L.

i k -Sublevel Cuts. i k -sublevel cuts of i-level cut A [i]
are obtained by frst specifying the index i � m, then further specifying the index k, and varying only j. Tese sub-level cuts are rows of an N × L submatrix as shown in equation:

i k j -Sub-Sublevel Cuts. i k j -sub-sublevel cuts A
constructed by frst specifying i and then further specifying both j and k. Te mathematical expression of these subsublevel cuts is given as follows: Note: it is obvious that A [i j k ] � [μ A l n (x m )] represents a single element of the expanded matrix and serves as a zoomin view of the matrix. For example, A [1 1 1 ] is an i k j -subsublevel cut that focuses on the information of the frst subject for its frst attribute at the frst time level.
i k j sub-sublevel cut: for a given i-level cut by specifying j, one can obtain the i k j -sub-sublevel cut.
It is obvious that by following the abovementioned split procedure, one would zoom into the given PCTLHS matrix. Te level cutbehaves like a zoom into the frst layer (matrix) of the hypermatrix, then the sub level cut reaches the the column or row of matrix, and the sub sub level cut would be an element of the column or row. Similarly, the reverse process can serve as a zoom-out function. In this way, one can approach the smallest unit of the extended universe, which is an explicit view of the event as an element of the PCHS matrix. Similarly, the matrix itself serves as an implicit view of the event.  symptoms were observed in their two diferent visits to the doctor. Tese two visits at two distinct times are considered two-time levels. In both these times, the symptoms (time-lined subattributes) of these patients were observed and recorded during the frst and second meetings with the doctor. Te three patients under observation are considered subjects. Te information/data of their health condition associated with both visits are organized in a PCTLHS matrix. Te clinical observations of the visits are expressed in the crisp environment and analyzed by using the plithogenic crisp hypersoft matrix.

Application
Consider the set of the frst three patients who are under observation.
Let the four attributes be A k j ;j � 1, 2, 3, 4 observed at two-time levels k � 1, 2 which are described as follows.
Fever with numeric values, k � 1, 2 representing frst and second-time levels  2 4 � Sickness state at the second visit Now, in the next two subsections, this information consisted of symptoms (attributes) of patients (subjects) observed at two levels of time. Two forms of expressions are formulated to organize information. One way of representation is the set expression, i.e., the PCTLHS set, and the other way of representation is the matrix expression, which is a connected matrix or a hypermatrix termed the PCTLHS matrix.

PCTLHS Set Representation.
Let the function A refects given attributes/time-leveled attributes described as follows: 4 is a combination of attributes at the frst visit level (α-combination). A 2 1 , A 2 2 , A 2 3 , A 2 4 is a combination of attributes at the second visit level (β-combination).
Individual crisp memberships are assigned to A � x 1 , x 2 , x 3 according to the opinion of the doctor, and then, information is represented as the PCTLHS set by using crisp memberships; that is, if a given symptom is present in the patient, the assigned membership is one, and if it is not present, the membership is zero. Te formal notation of the PCTLHS set A is described as follows: where μ A k j (x i ) represents crisp memberships assigned to three subjects (patients), i � 1, 2, 3 according to four different attributes (symptoms) j � 1, 2, 3, 4 that are observed at two distinct time levels k � 1, 2, (these plithogenic crisp memberships refect whether the A k j attribute is present . Te information of the frst visit of patients x i associated with four symptoms as α-combination of attributes is organized as a PCTLHS set: (1, 0, 1, 1), 1, 1, 1), 0, 0, 1).
It is clear that the frst-level layer of the PCTLHS matrix is generated by this A(α). Now, regarding the second observation of patients as β -combination of attributes, information is portrayed as a PCTLHS set: 0, 1, 0).
A(β) generates the second-level layer of the matrix.

PCTLHS Matrix Representation. Let
A is the matrix of representation for both PCTLHS Sets. Here, rows of this matrix represent x 1 , x 2 , x 3 (physical bodies or subjects) and columns represent (the nonphysical aspect of subjects as symptoms) attributes A k 1 , A k 2 , A k 3 , A k 4 . Tis whole information of two sets A(α) and A(β) is organized as two connected layers of PCTLHS matrix A described as follows: Tis PCTLHS matrix consists of two layers. Te frst layer is interpreted as the frst observation, representing the state of health of three patients. By observing the matrix given in equation (19), one can clearly see that, on the frst visit, the patient x 1 is sufering from fever with no dry cough but feeling sufocation and nausea. Te patient x 2 has a fever with dry cough, sufocation, and nausea. Te patient x 3 is sufering from fever with dry cough, no breathing difculty, and feeling nausea. However, we can observe from matrix equation (19) that, on the second visit, all symptoms are cleared in the x 1 patient, while the x 2 patient is having breathing difculty, and hence, the x 2 patient is sufering from fever and breathing Applied Computational Intelligence and Soft Computing difculty both. In this way, one can see and classify all information at a glance. Terefore, it is obvious that the matrix expression is the most appropriate expression to represent multidimensional data compared to the classical set expression.
A � μ A k j x i , i � 1, 2, 3, j � 1, 2, 3, 4, and k � 1, 2. (20) Te given PCTLHS matrix A is a rank-three tensor. Te Te top-to-bottom view of this PCTLHS Matrix consists of three parallel layers of ordinary matrices, and the order of each matrix is 2 × 4. Tese top-to-bottom layers in the separated form are i-level cuts.

i-Level Cuts A [i] of the PCTLHS-Matrix (Subjectwise Level Cuts)
A � Equation (21) represents a top-to-bottom view of the PCTLHS matrix, which is used to formulate i-level cuts A [i] (subjectwise level cuts) of the PCTLHS matrix.
We split the matrix at i � 1, 2, 3. Tese three integer levels (i splits) are three i-level cuts known as subjectwise level splits. Tese level cuts focus on subjects initially as follows: A [1] refects the attributive state of the frst patient x 1 observed in the two given time levels.
A [2] refects the attributive state of the second patientx 2 observed in the two given time levels.
A [3] refects the attributive state of the third patient x 3 observed in the two given time levels.
Tis matrix form represents three top-to-bottom layers, and each layer is a matrix of order 2 × 4. Tese layers are obtained by splitting the given matrix on the index i � 1, i � 2, i � 3. Each i-level cut represents states of four attributes (symptoms) associated with a specifc subject (patient) for the frst and second examination levels. Tese three i-level cuts focus on the patient frst and then describe their health state at time levels. Tese levels are utilized to organize the information for each patient separately. For example, the frst layer (i-level cut) of the above matrix refects the state of the patient x 1 .
From the frst layer, it can be seen that they have all three symptoms except for cough on the frst visit. During the second visit, all symptoms disappeared. Similarly, the second patient x 2 has all symptoms in the initial stage, and only one (cough) is left in the next stage. A description for the third patient can be given similarly by observing the third-level layer.

i j -Sublevel Cuts.
For specifed i � 1 and then further specify each j � 1, 2, 3, 4, respectively, i j -sublevel cuts A [i j ] of the i-level cut A [1] are constructed as follows: A [1 1 ] represents the frst symptom of the frst patient in two given time levels, which means after focusing on the patient, the next focus is on the attribute. Similarly, one can describe other two level cuts A [1 2 ] , A [1 3 ] and i j -sublevel cuts.

i j k -Sub-Sublevel
Cut. i j k -sub-sublevel cuts are constructed as described. By using a level cut, one would observe the patient frst i � m, j � n and k � l, then at the 8 Applied Computational Intelligence and Soft Computing next level by using a sublevel cut, one would observe his symptom (attribute) at several given time levels. And by using sub-sub-level cut, one would observe its symptom at a certain time level. Te sub-sublevel cut is the smallest unit of the matrix, which is the single element. Te general form is described as follows: i j k sub-sublevel cuts of A are as follows: ,

i k -Sublevel Cuts.
For given i-level cuts, further specifying k and varying only j, one can obtain i k -sublevel cuts. Tese level cuts are rows of an N × L submatrix. Specifying i � mk � l and varying j � 1, 2, . . . , Ni k -sublevel cuts for a fxed i-level cut are constructed as follows: i k -sublevel cuts of i-level cut A [i] are as follows: For fxed i � 2 and k � 1, 2,i k -sublevel cuts of i-level cut A [2] are as follows: For fxed i � 3 and k � 1, 2,i k -sublevel cuts of i-level cut A [3] are as follows:

i k j -Sub-Sublevel
Cut. i k j sub-sublevel cuts are constructed by specifying i � m. Ten, by further specifying j � n and k � l, the general form is described as follows: i j k sub-sublevel cuts of A are as follows:

Applied Computational Intelligence and Soft Computing
Te matrix in equation (36) is a 3 × 3 × 2 PCTLHS matrix with three subjects, three attributes, and two time levels.
i − level cuts of B are given as follows: Te local aggregation operators for i-level cuts are the union, intersection, average, and compliment operators described as follows. [i] is defned as follows:

Union of i − Level Cuts. Te union of A
[Ω A k j (X)] is an accumulated layer (submatrix) of the highest memberships considered as the top-level layer.

Example 2. For matrix
Te cumulative attributive state of all three subjects refects the highest level of each symptom among the given group of patients.

Intersection of i − Level Cuts. Te intersection of A [i]
defned as follows: [μ A k j (x)] is the accumulated lowest membership as a bottom layer that represents the lowest state of each symptom of the group of three patients.
Tis is the accumulated lowest attributive state of all three subjects as a group.

5.2.7.
Average of i − Level Cuts. Te average memberships as an interior layer are defned as follows: is the accumulated layer of average memberships considered as the interior-level layer. Tis average operator presents the neutral or average state of the universe.
Tis is the accumulative attributive state of all three subjects as they are viewed as a single entity, refecting the average level of each symptom.

Complement of i − Level Cuts.
Te complement of each membership of the i-level cut is defned as follows: Example 5. Complement of i-level cuts of A is given as follows: C A [1] � 0 1 0 0 Tese complements can be further accumulated through previously defned aggregation operators. Tese complement operators refect the inverted state of the merged universe.

Conclusion and
Discussion. Tis novice model would help improve and expand the feld of decision-making and artifcial intelligence. Using the PCTLHS matrix, one would be able to represent an extensive, indeterminate plithogenic universe. Some important facts of this model are discussed as follows: (1) Te PCTLHS matrix provides multidimensional views of the universe (subjects versus attributes and timelined attributes), whereas in this frst version, only one of the three possible views is described as a model. In this view, the subjects are initially focused by the observer, and the next focus would be attributes or time levels depending on the interest of the observer or decision-maker. (2) One can classify and analyze the universe explicitly and implicitly through level cuts, sublevel cuts, and subsublevel cuts. In this way, one could be able to enhance his approach to knowing the universe (all information) in a more dynamic and modern form. (3) By choosing specifc i, j, or k level cuts, one can make his priority choices by subjects, attributes, or time levels, and by specifying further sublevel cuts, one can further specify his next level of selection by subject, attribute, or time levels. Trough the next level of selection, i.e., using sub-sublevel cuts, one can approach a fnal selection of the subject, attribute, or time level as per requirements. (4) Te PCTLHS matrix provides the broader exterior and interior views of the matrix by displaying all possible events (realities) together. Terefore, we expect that this PCTLHS Matrix will deal with multiple information networks under one command and one control system. (5) Te level cuts of the PCTLHS matrix present an explicit event or reality at a single instance of time. Terefore, one would be able to observe the state of an event (health state of patient) from moment to moment. Tis will improve his understanding of each event and every state according to its time level during that state. (6) Variation index-based level cuts provide the view of reality or events from multiple angles of vision. Tis will improve the mathematical visualization of facts/ events/information. (7) One can analyze the universe by choosing the best possible reality from several possible realities using level cuts and operators. Tis fact would be helpful in the development of artifcial intelligence programs. (8) Tis PCTLHS matrix is very close to the functions of the human mind. Terefore, it acts as a pioneer in the modeling of the human mind. If this application can be opted by some computer specialists/software developers, it has every potential not only to improve artifcial intelligence (AI) systems but also has great promises in the feld of real intelligence (RI). (9) Te disjunction operator, i.e., the max operator, provides the optimist view of reality. (10) Te conjunction operator, i.e., the min operator, ofers the pessimist perception of the event or reality (11) Te averaging operator ofers the neutral states of the unifed universe, which means that all the elements of the universe are considered as a single entity and only its neutral states are focused (12) Te complement operator presents the inverted refection of the event or reality. (13) Te local operators designed for i-level cuts serve the purpose of unifying the matter bodies of the universe. Tis means that all the elements of the universe are merged and presented as a single matter body refecting several attributes in distinct time levels. In this way, the concept of the unifed global matter (something like dark matter) is visualized. (14) Tese local operators serve the purpose of unifcation of matter bodies of the universe. Tis means that all the elements of the universe are merged and presented as a single matter body refecting several attributes in distinct time levels. Tis is how the concept of the unifed global matter (something like dark matter) arises. In this way, the concept of the unifed global matter (something like dark matter) emerges.

Comparisons of Former Fuzzy Extensions and Models.
Tis section contains a brief comparison of previous and current fuzzy extensions and models. Te soft set is an advanced and extended version of the fuzzy set because it manages many attributes simultaneously regardless of the fuzzy set, which only handles one attribute at a time.

Applied Computational Intelligence and Soft Computing
Te hypersoft set is an enhanced extension of a soft set because it can accommodate multidimensional information by managing many attributes and their various values simultaneously.
Te plithogenic hypersoft set is a more advanced and applicable version than the hypersoft set, soft set, and fuzzy set. It is a higher dimensional version that manages detailed information. Te observer can intrinsically see the state of the element x (subject) by looking at each attribute separately. In simple terms, the plithogenic hypersoft set manages multiple attributes and their values (subattributes) simultaneously and beyond by observing each attribute separately.
Te plithogenic fuzzy whole hypersoft set/matrix (PFWHS set/matrix) is a more appropriate choice than the previously mentioned extensions as it indicates the states of subjects (attributes/subattributes) at the individual level for each attribute/subattribute (as in the case of the plithogenic hypersoft set) and also at the unifed or combined level for all attributes together as a whole (the case of the hypersoft set). Terefore, it is an extended hybrid version of the hypersoft set and the plithogenic hypersoft set. By using the PFWHS set/matrix, one can observe a more transparent inner perception (case of a single state representation) or outer view (case of a combined state representation) of information/facts/events.
Te plithogenic subjective hyper-supersoft matrix (PSHSS matrix) is a generalized advanced form of the PFWHS matrix and is more applicable as than previously developed models, as it has a greater capacity to express and manage various connected attributes/subattributes separately and as a whole by considering connected attribute/ subattribute levels.
Te plithogenic time-leveled hypersoft matrix (PTLHS matrix) is a special case of the previous generalized form (PSHSS matrix) that manages time-based connected attributes. It is more suitable than other extended fuzzy sets mentioned (soft set, hypersoft set, plithogenic hypersoft set, PFWHS set/ matrix, and PSHSS matrix) for the following valid reasons: (1) Te plithogenic time-leveled hypersoft matrix includes time as the fundamental source of variation.
As most of the variations in this universe are timedependent like weather graphs, stock exchange, and website ratings, therefore, it is of tremendous help if this PCTLHS matrix is used to cope with the scattered time-varying piece of information. (2) It deals with several attributes interiorly such that each attribute has many values (subattributes). Tese subattributes are varying with the fow of time. By using the PCTLHS matrix, one can organize and classify multidimensional information into the shape of connected matrix layers as hypermatrices. (3) Te matrix expression is the most appropriate expression to represent multidimensional data compared to the classic set expression. (4) Tis PTLHS Matrix allows the viewer to see information down to its innermost level through level cuts, sublevel cuts, and sub-sublevel cuts.
(5) In addition, it ofers a broader view of multidimensional information by viewing the entire universe as a hypersoft time-leveled matrix. Terefore, the observer can see and analyze the whole universe externally at a single glance. (6) Level cuts can focus on one required piece of information that is displayed in the form of a single matrix layer of the PTLHS matrix, while the other information can vary with the fow of time being displayed as other matrix layers. (7) Sublevel cuts can focus on required information that is displayed as a single column or row of the given layer (submatrix) of the PTLHS matrix. (8) Sub-sublevel cuts can focus on required information that is displayed as a single element of the submatrix of the PTLHS matrix. (9) Sublevel cuts ofer the representation of the previous lower dimension in the further lower dimension and enable us to sneak in an inside view of the expanded universe; i.e., after explicitly focusing on a subject through an i-level cut (single level of the layered matrix), our next focus is on that subject's (patient's) attribute (a particular symptom) through the sublevel cut (row or column of one layer of the multilayered matrix). (10) It also ofers the unifcation of the information by applying aggregation operators, and in this way, all the extended information of the universe that is represented as a matrix having multiple layers can be transformed into a single layer of the matrix. (i) In this article, we have portrayed the plithognic hypersoft matrix in a crisp environment. One can extend this model in other environments such as fuzzy, intuitionistic, and neutrosophic, or any mixed or combined environment; i.e., containing several environments would provide the variation of fuzziness levels of refected events. (ii) Moreover, some other kinds of local operators can be provided for the unifcation purpose of littered data according to the requirement of the concerned bodies. (iii) Te operations and properties of these hypersoft matrices need to be explored.

Data Availability
No data were used to support the fndings of this study as it is a theoretical model.

Conflicts of Interest
Te authors declare that they have no conficts of interest.