Mathematical Models of Diagnostic Information Granules Generated by Scaling Intuitionistic Fuzzy Sets

Abstract

The paper presents a certain class of the mathematical models of diagnostic information granules describing the fuzzy symptoms-faults relationship. A certain fuzzy diagnostic information retrieval system is described as an application of an expert diagnostic system. Symptoms and faults are fuzzy, and with some scaling of the symptom-fault concept pair values. These value pairs can be considered as intuitionistic fuzzy sets for the space of diagnosed objects. In this article, for scaling intuitionistic fuzzy sets (n-ScIFS), the deductive theory is formulated. There the intuitionistic fuzzy sets (IFSs) and the Pythagorean fuzzy sets (PFSs) are generalized to the n-ScIFS objects. The membership and non-membership values, as standard, can be described by the 1:1 scale or the quadratic function scale. However, any power scale $n>2$ can be used. In this paper, any n-Sc scaling functions retaining the IFSs properties are considered. The n-ScIFS theory introduces a conceptual apparatus analogous to the classical theory of Zadeh fuzzy sets and Yager PFSs, consistently striving, for the first time, to formulate the relational structure of n-ScIFSs as a model of a certain information granule system called here the diagnostic granule system. In addition, power- and linear-repeatable diagnostic granules are defined in the n-ScIFSs structure for serial or parallel diagnosis processes. The information granule base is determined and a diagnostic granule system model produced by this information granule base is shown. Certain algorithms have been given to establish the semantic language of diagnosis describing the system of diagnostic information granules.

1. Introduction

The fuzzy sets introduced by Zadeh [1] are widely used in many different disciplines of science. They can be found in many solutions in engineering, management, mathematics, statistics, social sciences, cybersecurity, medicine, and others. Using fuzzy set theory is also an important issue in medical and technical diagnostics. There are several some important fields and problems of diagnostics where fuzzy sets are used:

• dynamic diagnosis: the issues of dynamic diagnosis, especially fault detection and isolation, are presented in many papers, books, publications, and monographs [2,3,4,5,6,7,8,9,10,11]. Commonly, when the complex industrial plants are diagnosed, there is a necessity to decompose this process into some smaller units. Then, these units can be simultaneously diagnosed [12,13];
• static diagnosis: the problems of static diagnosis concern signal analysis [14]; isolation of faults and the fault-symptom relations [15]; application of fuzzy logic in diagnostics [16]; using the diagnostic matrix to generalized reasoning about faults [17]; and diagnostic test using fault-symptom relations [18];
• mathematical models of information granules are being built [19,20]. They are models describing aggregated fuzzy diagnostic data [21,22] in the symptom-faults matrix. This type of information granules are also intuitionistic fuzzy sets presented in [23,24], as applications in management and engineering. The expert diagnostic systemis described as one that performs calculations on numerical data representing fuzzy diagnostic information. The application of this system are the procedures for searching for diagnostic information specified in the diagnosis granule systems. Thus, this paper presents some mathematical model of diagnostic information granules of the symptoms-faults relationship. Symptoms and faults are fuzzy. When some scale is used for values of the symptom-fault concept pair, then they can be considered as intuitionistic fuzzy sets in the space of diagnosed objects. In this purpose, the deductive theory of scaling intuitionistic fuzzy sets (n-ScIFS) is formulated.

Generally, diagnosticsexamines the symptoms of threats in obtaining information about the expected behavior of technical, medical, information-obtaining objects (in the web, in cyberspace), and others. One of the main goals of diagnostics is therefore to indicate the occurrence of threats f for the s states of the expected behavior of objects, i.e., to indicate their limitations and selection by occurrence of faults, defects, damage, or deficiencies that threaten the occurrence of the expected behavior of objects. The states s of the expected behavior of objects can be endangered in a real object and recognized on the basis of diagnostic signals generated by this object. Then these conditions are called threat symptoms. Of course, other external influences may also occur, however to simplify, they will be ignored in this study.

The feature of the state or symptom of an object is a component of the instantaneous state of the object, related to the property of the examined object. The symptom is some feature that characterize an object and the state of this object, which occur only when the object is not fully fit or damaged. Furthermore, the diagnostic symptom s defines the measure of the diagnostic signal and is called the signal vector component. This signal represents the specific type of fault or threat f.

The parameter is the feature that determines the condition of the examined object. Moreover, the diagnostic parameter indicates some diagnosed object, the observable description of this object, by some diagnostic signal or process. This signal indirectly represents some value of the state of the object.

In order to make a decision in the diagnostic system, the diagnostic inference is used. The response of this system, which is a decision, is based on the diagnostic symptom-state relationship. Symptoms are conditional attributes, while faults are decision attributes. Let the set $S=\{s_1,s_2,\dots ,s_l,\dots ,s_k\}$ be the set of symptoms attributes and let $F=\{f_1,f_2,\dots ,f_l,\dots ,f_n\}$ be the set of faults attributes.

Let the set $X=O\times S\times F$ be called the diagnostic range of objects o with the symptom $s_i$ and faults $f_j$ attributes. Some data about these objects are fuzzy, then the diagnosis of the object $x\in X$ with the attributes $s_i,f_j$, with some scaling, corresponds to the pair $({\mu }_i,v_j)$ of fuzzy sets ${\mu }_i:X\to [0,1],v_j:X\to [0,1]$. Observations of symptoms and faults are disjoint sets. The values of functions ${\mu }_i$ and $v_j$ satisfy the condition:

$${\mu }_i\left(o\right)+v_j\left(o\right)\le 1.$$

(1)

This means that the pair of fuzzy sets $({\mu }_i,v_j)$ is the intuitionistic fuzzy set (IFSs) in the Atanassov sense [25].

In the Atanassov study [25], the intuitionistic fuzzy sets (IFSs) consider that there is the value $\mu$ of the membership function and additionally there is the value v of the non-membership function. In this research, the following constraint is taken into account: $0\le \mu +v\le 1$. Moreover, in the Yager study [26,27], the Pythagorean fuzzy sets (PFSs) model is proposed in order to handle imprecise information. If we compare this model to the Atanassov one, the IFSs condition is relaxed and has the form: $0\le {\mu }^2+v^2\le 1$. In the literature, the research of Zhang and Xu [28] can be found. They established the model of the Pythagorean fuzzy number (PFN). Furthermore, Garg’s research [29,30] shows the applications of PFSs in decision-making problems.

The membership and non-membership values are usually measured for some scale function. Until now, they have been described on the scale 1:1 [31] or on the quadratic function scale of PFSs [32]. In [33], scales for the proven power $n>2$ are used. In [24], the function scale is the power function $Sc\left(x\right)=x^a,a\in (1,+inf))$.

In addition, in the IFS articles [31,34,35,36] and those related to PFSs [26,27,32,37,38,39,40], a number of mathematical and logical inaccuracies are corrected and presented in [33]. The paper [33] presented the elements of the deductive theory of n-PFS, which determines the membership degree $\mu$ and the non-membership degree v not only for the square scale, but also for any power scale, i.e.: $0\le {\mu }^n+v^n\le 1$, for some $\mu ,v\in [0,1]$, if $\mu +v>1$. This makes it possible to formulate any local deductions in terms of n-PFS.

In this context, the motivation for this paper was the use of the described model to create analogous models, however based on any selected functional scales, e.g.,

• The map scale $f_n\left(x\right)=x/n:0\le \mu /n+v/n\le 1$,
• Or the sequence of logarithmic scale functions $f_n\left(x\right)=1-{log}_{(1-1/n)}x$, for $n>1,x\in (0,1]$, where $0\le {log}_{(1-1/n)}\mu +{log}_{(1-1/n)}v\le 1$.

However, two research problems arose:

• How to determine the sequences $\left\{S{c}_n\right\},\left\{S{c}_n^{\prime }\right\}$ of scaling functions, such that from a certain number n there is $0\le S{c}_n\left(\mu \right)+S{c}_n^{\prime }\left(v\right)\le 1$, for any results of diagnostic measurements $\mu ,v\in [0,1]$,
• How to determine the number of n sequences $\left\{S{c}_n\right\},\left\{S{c}_n^{\prime }\right\}$ satisfying the condition 1.

In this paper, the solution to problems 1 and 2 is presented for the first time.

Furthermore, the conceptual apparatus describing n-PFSs has been generalized and clarified [33]. Furthermore, for the sequence $\left\{S{c}_n\right\}$ of some scaling functions, for any natural number $n>0$, deductive theories of n-scaling intuitionistic fuzzy sets, with the condition $0\le S{c}_n\left(\mu \right)+S{c}_n{\left(v\right)}^{\prime }\le 1$, are formulated. The constructions of these theories are presented below. Their formal description is called the diagnostic granule system.

The following are presented in the coming sections: the norm systems, the n-scaling intuitionistic fuzzy sets, Yager aggregation operators, and the triangular norms in n-ScIFN and n-ScIFN algebra. By intuition, it is assumed that this system can interpret a diagnostic descriptive language that describes the symptoms-faults properties. In the fifth chapter of this paper, it is shown that the n-scaling intuitionistic fuzzy sets theory is a model of the diagnostic granule system and that this system differs from the information granule system [20] in terms of including linear and power repetition operations on granules. The final chapter of this paper presents a simple example of a standard diagnostic granule system.

2. The Norm Systems

2.1. The Triangular Norms

Triangular norms are used to construct operations on fuzzy sets and Yager [32] fuzzy sets.

Definition 1.

The triangular norm or thet-normin $[0,1]$ is the operation ${•}_t:[0,1]\times [0,1]\to [0,1]$, when for any numbers $x,y,z\in [0,1]$ meets the following conditions:

• $0{•}_{t}y=0,y{•}_{t}1=y$—boundary conditions,
• $x{•}_{t}y\le z{•}_{t}y$, when $x\le z$—monotonicity,
• $x{•}_{t}y=y{•}_{t}x$—commutativity,
• $x{•}_t\left(y{•}_{t}z\right)=\left(x{•}_{t}y\right){•}_{t}z$—associativity.

The triangular t-conorm or thes-normin $[0,1]$ is the operation ${•}_s:[0,1]\times [0,1]\to [0,1]$, when for any numbers $x,y,z\in [0,1]$ it meets the following conditions:

• $0{•}_{s}y=y,y{•}_{s}1=1$—boundary conditions,
• $x{•}_{s}y\le z{•}_{s}y$, when $x\le z$—monotonicity,
• $x{•}_{s}y=y{•}_{s}x$—commutativity,
• $x{•}_s\left(y{•}_{s}z\right)=\left(x{•}_{s}y\right){•}_{s}z$—associativity.

Theorem 1.

Operation $•:[0,1]\times [0,1]\to [0,1]$ is the s-norm if it is defined by the formula $x•y=1-(1-x){•}_t(1-y)$, for any numbers $x,y\in [0,1]$.

Example 1.

The operation defined by the formula $x{•}_{t}y=min\{x,y\}$ for any $x,y\in [0,1]$ is a t-norm.

The operation defined by the formula $x{•}_{s}y=max\{x,y\}$ for any $x,y\in [0,1]$ is the s-norm.

Further in this paper, we consider only the continuous t-norms and s-norms. The general construction of triangular norms and the functional equations [41] leads to the Theorem:

Theorem 2.

For each strictly decreasing continuous function $f_t:[0,1]\to [0,+\infty )$ such that $f_t\left(1\right)=0$ and for any $x,y\in [0,1]$, the t-norm is the operation determined by the formula:

$$x{•}_{t}y=\left\{\begin{array}{c}{f}_t^{-1}[f_t\left(x\right)+f_t\left(y\right)],for{f}_t\left(x\right)+f_t\left(y\right)\in [0,f_t\left(0\right)],\hfill \\ 0,otherwise.\hfill \end{array}\right.$$

(2)

For each strictly increasing continuous function $f_s:[0,1]\to [0,+\infty )$ such that $f_s\left(0\right)=0$ and for any $x,y\in [0,1]$, the s-norm is the operation determined by the formula:

$$x{•}_{s}y=\left\{\begin{array}{c}{f}_s^{-1}[f_s\left(x\right)+f_s\left(y\right)],\mathrm{for}{f}_s\left(x\right)+f_s\left(y\right)\in [0,f_s\left(1\right)],\hfill \\ 1,otherwise.\hfill \end{array}\right.$$

(3)

Then the $f_t,f_s$ functions are called t-norm and s-norm generators, respectively.

When the functions $f_t,f_s$ satisfy the condition: $f_s\left(x\right)=1-f_t(1-x)$ for every $x\in [0,1]$, then they are called dual t-norm and s-norm generators.

Remark 1.

Only dual standard generators will be described hereinafter.

Example 2.

For any $x,y\in [0,1]$, if the t-norm $x{•}_{t}y=1-min\{1,{({(1-x)}^p+{(1-y)}^p)}^{1/p},p\ge 1$ is considered, then the generator is $f_t\left(x\right)=1-x^p$.

For any $x,y\in [0,1]$, if the s-norm $x{•}_{s}y=min\{1,{(x^p+y^p)}^{1/p}\},p\ge 1$ is considered, then the generator is $f_s\left(x\right)=x^p$.

2.2. The Repeatability Norms

For any generators of triangular norms $f_t,f_s$, and any number $x\in [0,1]$, assuming notation $x=x_1=x_2=...=x_k$:

$y=x_1{•}_t{x}_2{•}_t...{•}_t{x}_k=f_t^{-1}\left[k{f}_t\left(x\right)\right]$ for $k{f}_t\left(x\right)\in [0,f_t\left(0\right)],$

$z=x_1{•}_s{x}_2{•}_s...{•}_s{x}_k=f_s^{-1}\left[k{f}_s\left(x\right)\right]$ for $k{f}_s\left(x\right)\in [0,f_s\left(1\right)].$

Thus, the k-times execution of the triangular norm for the number x is a number which generator value is k-times the generator value for the number $x:f_t\left(x\right)=(1/k)f_t\left(y\right)$ and $f_s\left(x\right)=(1/k)f_s\left(z\right)$.

Hence $i{f}_t\left(x\right)=(i/k)f_t\left(y\right)$ and $i{f}_s\left(x\right)=(i/k)f_s\left(z\right)$. Moreover, for $i<k$:

$x_1{•}_t{x}_2{•}_t...{•}_t{x}_i=f_t^{-1}\left[(i/k)f_t\left(y\right)\right],$

$x_1{•}_s{x}_2{•}_s...{•}_s{x}_i=f_s^{-1}\left[(i/k)f_s\left(y\right)\right].$

Since for any $y\in [0,1]$ there exists a number $x\in [0,1]$ satisfying the above formulas, we can assume the notation:

$(i/k){•}_{p}y{=}_{df}{f}_t^{-1}\left[(i/k)f_t\left(y\right)\right],$

$(i/k){•}_{l}z{=}_{df}{f}_s^{-1}\left[(i/k)f_s\left(z\right)\right].$

If we assume that the procedures of repeating norms (calculating the values of the generators of these norms) for any x can be used only with a certain degree $\lambda \in [0,1]$, it will be natural to adopt following Definitions of repeatability norms by applying these procedures.

Definition 2.

The operation ${•}_p:[0,1]\times [0,1]\to [0,1]$ will be called the power repeatability standardof the t-norm ${•}_t:[0,1]\times [0,1]\to [0,1]$ (briefly: p-norm), if the following conditions are met, for any ${\lambda }_1,{\lambda }_2,x,y\in [0,1]$:

• $1{•}_{p}x=x,0{•}_{p}x=1$,
• if $x\ne 0$ and ${\lambda }_1{•}_{p}x={\lambda }_2{•}_{p}x$, then ${\lambda }_1={\lambda }_2$,
• if $x\ne 0,y\ne 0,{\lambda }_2\ne 0$, and ${\lambda }_1{•}_{p}x={\lambda }_2{•}_{p}y$, then $({\lambda }_1/{\lambda }_2){•}_{p}x=y,$
• $({\lambda }_1\ast {\lambda }_2){•}_{p}x={\lambda }_1{•}_p\left({\lambda }_2{•}_{p}x\right),$
• $({\lambda }_1+{\lambda }_2){•}_{p}x=\left({\lambda }_1{•}_{p}x\right){•}_t\left({\lambda }_2{•}_{p}x\right),$
• ${\lambda }_1{•}_p\left(x{•}_{t}y\right)=\left({\lambda }_1{•}_{p}x\right){•}_t\left({\lambda }_1{•}_{p}y\right).$

The operation ${•}_l:[0,1]\times [0,1]\to [0,1]$ will be called the linear repeatability standardof the s-norm ${•}_s:[0,1]\times [0,1]\to [0,1]$ (briefly: l-norm), if the following conditions are met, for any ${\lambda }_1,{\lambda }_2,x,y\in [0,1]$:

• $1{•}_{l}x=x,0{•}_{l}x=0,$
• if $x\ne 0$ and ${\lambda }_1{•}_{l}x={\lambda }_2{•}_{l}x$, then ${\lambda }_1={\lambda }_2,$
• if $x\ne 0,y\ne 0,{\lambda }_2\ne 0$, and ${\lambda }_1{•}_{l}x={\lambda }_2{•}_{l}y,$ then $({\lambda }_1/{\lambda }_2){•}_{l}x=y,$
• $({\lambda }_1\ast {\lambda }_2){•}_{l}x={\lambda }_1{•}_l\left({\lambda }_2{•}_{l}x\right),$
• $({\lambda }_1+{\lambda }_2){•}_{l}x=\left({\lambda }_1{•}_{l}x\right){•}_s\left({\lambda }_2{•}_{l}x\right),$
• ${\lambda }_1{•}_l\left(x{•}_{s}y\right)=\left({\lambda }_1{•}_{l}x\right){•}_s\left({\lambda }_1{•}_{l}y\right).$

Definition 3.

The system ${\mathbf{S}}_{\mathbf{Yager}}=\langle [0,1],{•}_t,{•}_s,{•}_p,{•}_l,0,1\rangle$ is called the Yager system of norms (triangular and repeatability).

Theorem 3.

Let $f_t$ be a generator of the triangular norm ${•}_t$. Then there are operations ${•}_p:[0,1]\times [0,1]\to [0,1]$ defined by formula:

$$\lambda {•}_{p}x=\left\{\begin{array}{c}{f}_t^{-1}\left[\lambda f_t\left(x\right)\right]\mathrm{for}\lambda f_t\left(x\right)\in [0,f_t\left(0\right)],\hfill \\ 0,otherwise.\hfill \end{array}\right.$$

(4)

Let $f_s$ be a generator of the triangular norm ${•}_s$. Then there are operations ${•}_l:[0,1]\times [0,1]\to [0,1]$ defined by formula:

$$\lambda {•}_{l}x=\left\{\begin{array}{c}{f}_s^{-1}\left[\lambda f_s\left(x\right)\right]\mathrm{for}\lambda f_s\left(x\right)\in [0,f_s\left(1\right)],\hfill \\ 1,otherwise.\hfill \end{array}\right.$$

(5)

Proof. 

Since $f_t$ is a strictly descending function, there is only one value of $f_t^{-1}\left[\lambda f_t\left(x\right)\right]\in [0,1]$, when $\lambda f_t\left(x\right)\in [0,f_t\left(0\right)]$. Let us denote that $\lambda {•}_{p}x{=}_{df}{f}_t^{-1}\left[\lambda f_t\left(x\right)\right]$, for $\lambda f_t\left(x\right)\in [0,f_t\left(0\right)]$, and $\lambda {•}_{p}x{=}_{df}0$, otherwise. We define the operation ${•}_l$ in the same way.    □

The following notation agreement is assumed here:

$\lambda {•}_{p}x{=}_{df}{x}^{\lambda }$ and $\lambda {•}_{l}x{=}_{df}\lambda x$.

Fact 1.

If $x^{\lambda }=f_t^{-1}\left[\lambda f_t\left(x\right)\right],$ for $\lambda f_t\left(x\right)\in [0,f_t\left(0\right)]$, then $f_t\left(x^{\lambda }\right)=\lambda f_t\left(x\right)$.

If $\lambda x=f_s^{-1}\left[\lambda f_s\left(x\right)\right]$, for $\lambda f_s\left(x\right)\in [0,f_s\left(1\right)]$, then $f_s\left(\lambda x\right)=\lambda f_s\left(x\right)$.

Theorem 4.

The repetition operations ${•}_p,{•}_l$ for Theorem 3 satisfy the following conditions, for any $x,y,\lambda ,{\lambda }_1,{\lambda }_2\in [0,1]$:

1. 

$\lambda \left(x{•}_{s}y\right)=\lambda x{•}_s\lambda y$,

2. 

${\left(x{•}_{t}y\right)}^{\lambda }=x^{\lambda }{•}_t{y}^{\lambda }$,

3. 

$({\lambda }_1+{\lambda }_2)x={\lambda }_{1}x{•}_s{\lambda }_{2}x$,

4. 

$x^{{\lambda }_1+{\lambda }_2}=x_1^{\lambda }{•}_t{x}_2^{\lambda }.$

Thus, they are p-norm and l-norm.

Proof. 

1.

$\lambda \left(x{•}_{s}y\right)=f_s^{-1}\left[\lambda f_s\left(x{•}_{s}y\right)\right]=f_s^{-1}[\lambda f_s\left(x\right)+\lambda f_s\left(y\right)]=f_s^{-1}[f_s\left(\lambda x\right)+f_s\left(\lambda y\right)]=\lambda x{•}_s\lambda y$, for $f_s\left(x{•}_{s}y\right)\in [0,1]$, and because $\lambda f_s\left(x{•}_{s}y\right)\le f_s\left(x{•}_{s}y\right),$ so $\lambda f_s\left(x{•}_{s}y\right)\in [0,1].$

2.

${\left(x{•}_{t}y\right)}^{\lambda }=f_t^{-1}\left[\lambda f_t\left(x{•}_{t}y\right)\right]=f_t^{-1}[\lambda f_t\left(x\right)+\lambda f_t\left(y\right)]=f_t^{-1}[f_t\left(x^{\lambda }\right)+f_t\left(y^{\lambda }\right)]=x^{\lambda }{•}_t{y}^{\lambda }$, for $f_t\left(x{•}_{t}y\right)\in [0,1]$, and because $\lambda f_t\left(x{•}_{t}y\right)\le f_t\left(x{•}_{t}y\right)$, so $\lambda f_t\left(x{•}_{t}y\right)\in [0,1].$

3.

$({\lambda }_1+{\lambda }_2)x=f_s^{-1}\left[({\lambda }_1+{\lambda }_2)f_s\left(x\right)\right]=f_s^{-1}[{\lambda }_1{f}_s\left(x\right)+{\lambda }_2{f}_s\left(x\right)]=f_s^{-1}[f_s\left({\lambda }_{1}x\right)+f_s\left({\lambda }_{2}x\right)]={\lambda }_{1}x{•}_s{\lambda }_{2}x$, for $({\lambda }_1+{\lambda }_2)f_s\left(x\right)=f_s\left({\lambda }_{1}x\right)+f_s\left({\lambda }_{2}x\right)\in [0,1]$.

4.

$x^{{\lambda }_1+{\lambda }_2}=f_t^{-1}\left[({\lambda }_1+{\lambda }_2)f_t\left(x\right)\right]=f_t^{-1}[{\lambda }_1{f}_t\left(x\right)+{\lambda }_2{f}_t\left(x\right)]=f_t^{-1}[f_t\left(x^{{\lambda }_1}\right)+f_t\left(x^{{\lambda }_2}\right)]=x^{{\lambda }_1}{•}_t{x}^{{\lambda }_2}$, for $({\lambda }_1+{\lambda }_2)f_t\left(x\right)=f_t\left(x^{{\lambda }_1}\right)+f_t\left(x^{{\lambda }_2}\right)\in [0,1].$

   □

3. The n-Scaling Intuitionistic Fuzzy Sets and Yager Aggregation Operators

In this section, a mathematical model of information granulation is presented. It is about the set X of the information instances obtained by measurements using a certain sequence of scaling functions $\left\{S{c}_n\right\}$.

Definition 4.

The sequence $\left\{S{c}_n\right\}$ of the function $S{c}_n:[0,1]\to [0,1]$ will be called the scale function sequence of the intuitionistic fuzzy sets when the following conditions are met:

1. 

$S{c}_1$ is a function of identity, $S{c}_n$ is a continuous and strictly increasing function in the interval $[0,1]$, for $n>1$,

2. 

$S{c}_n\left(0\right)=0,S{c}_n\left(1\right)=1$, for any n,

3. 

$S{c}_{n+1}\left(x\right)\le S{c}_n\left(x\right)$, for any n and $x\in [0,1]$,

4. 

For any numbers $u.v\in [0,1]$, there exists such function $S{c}_k$, that $0\le S{c}_k\left(u\right)+S{c}_k\left(v\right)\le 1$.

Example 3.

The map scale.A sequence of map functions $S{c}_n\left(x\right)=x/n$, for $n>0,x\in [0,1]$ is a sequence of scaling functions, since:

1. 

$S{c}_1\left(x\right)=x$ and functions $S{c}_n$ are strictly growing,

2. 

$S{c}_n\left(0\right)=0,S{c}_n\left(1\right)=1$, for any n,

3. 

$x/(n+1)\le x/n$ iff $S{c}_{n+1}\left(x\right)\le S{c}_n\left(x\right)$, for any n and $x\in [0,1]$,

4. 

For $k=2,u,v\in [0,1]$: $0\le u/2+v/2\le 1$,

5. 

$S{c}_1^{-1}\left(x\right)=x$ and for $n>1$: $S{c}_n^{-1}\left(x\right)=nx$.

Example 4.

The logarithmic scale.A sequence of logarithmic functions $S{c}_n\left(x\right)=1-{log}_{(1-1/n)}x$, for $n>1,x\in (0,1]$, when $S{c}_1\left(x\right)=x$ and $S{c}_n\left(0\right)=0$, is a sequence of scaling functions, since:

1. 

$S{c}_1\left(x\right)=x$ and functions $S{c}_n$ are strictly growing,

2. 

$S{c}_n\left(0\right)=0,S{c}_n\left(1\right)=1-{log}_{(1-1/n)}1=1$, for any $n>1$,

3. 

For any $x\in (0,1],x\le 1$ iff $1/2\le 1/2x$ iff ${log}_{1/2}1/2x\le {log}_{1/2}1/2$ iff $1-{log}_{1/2}x\le 1$ iff $S{c}_2\left(x\right)\le S{c}_1\left(x\right)$,

for any $n>1,x\in (0,1],$ $1-1/n\le 1-1/(n+1)$ iff ${log}_{(1-1/n)}x\le {log}_{(1-1/(n+1\left)\right)}x$ iff $1-{log}_{(1-1/(n+1\left)\right)}x\le 1-{log}_{(1-1/n)}x$ iff $S{c}_{n+1}\left(x\right)\le S{c}_n\left(x\right)$,

4. 

For $u,v\in [0,1]$, when $uv=0$ or for $uv=1$ there is $0\le S{c}_1\left(u\right)+S{c}_1\left(v\right)\le 1$,

for $uv<0$ there is such $k>1$, that $uv\le 1-1/k$ iff ${log}_{(1-1/k)}(1-1/k)\le {log}_{(1-1/k)}uv$ iff $2-{log}_{(1-1/k)}uv\le 1$ iff $(1-{log}_{(1-1/k)}u)+(1-{log}_{(1-1/k)}v)$ iff $0\le S{c}_k\left(u\right)+S{c}_k\left(v\right)\le 1$,

5. 

$S{c}_1^{-1}\left(x\right)=x$ and for $n>1$, $S{c}_n^{-1}\left(x\right)={(1-1/n)}^x.$

Example 5.

The power scaleA sequence of power functions $S{c}_n\left(x\right)=x^n$, for $n>0,x\in [0,1]$ is a sequence of scaling functions, since:

1. 

$S{c}_1\left(x\right)=x$ and functions $S{c}_n$ are strictly growing,

2. 

$S{c}_n\left(0\right)=0,S{c}_n\left(1\right)=1$, for any n,

3. 

$x\le 1$ iff $x^{(n+1)}\le x^n$ iff $S{c}_{n+1}\left(x\right)\le S{c}_n\left(x\right)$, for any n and $x\in [0,1],$

4. 

For $u,v\in [0,1]$ exists such $k1,k2$, that $u^{k1}\le 1/2$ and $v^{k2}\le 1/2$, and when $k=max\{k1,k2\}$, then $0\le u^k+v^k\le 1$, hence $0\le S{c}_k\left(u\right)+S{c}_k\left(v\right)\le 1$,

5. 

For $n>0$, $S{c}_n^{-1}\left(x\right)=x^{1/n}.$

Definition 5.

Let $F=\{f:X\to [0,1\left]\right\}$ be the set of all fuzzy sets on non-empty space X.

Any function $p:X\to [0,1]\times [0,1]$, for any ${\mu }_p,v_p\in F$ and some natural number $n>0$, is defined by formulas:

${\mu }_p=\{\langle x,u\rangle :$ exists $v\in [0,1]$ such that $p\left(x\right)=\langle u,v\rangle ,x\in X\},$

$v_p=\{\langle x,v\rangle :$ exists $u\in [0,1]$ such that $p\left(x\right)=\langle u,v\rangle ,x\in X\},$

$p=\{\langle x,\langle {\mu }_p\left(x\right),v_p\left(x\right)\rangle \rangle :x\in X\},$

will be called then-scaling intuitionistic fuzzy sets(n-ScIFS) for sequences $\left\{Scn\right\},\left\{Sc{n}^{\prime }\right\}$ of scaling functions, if the condition is met:

$0\le S{c}_n\left({\mu }_p\left(x\right)\right)+S{c}_n^{\prime }\left(v_p\left(x\right)\right)\le 1$, for any $x\in X.$

The n-ScIFS is a set of all n-scaling intuitionistic fuzzy sets for sequences $\left\{Scn\right\},\left\{Sc{n}^{\prime }\right\}$ of scaling functions.

The fuzzy set ${\mu }_p$ indicates the membership and $v_p$ indicates the non-membership.

Let $p{=}_{df}\langle {\mu }_p,v_p\rangle$ represent the n-fuzzy number n-ScIFN. Then $p\left(x\right){=}_{df}\langle {\mu }_p\left(x\right),v_p\left(x\right)\rangle$ and $\langle {\mu }_p,v_p\rangle \left(x\right){=}_{df}\langle {\mu }_p\left(x\right),v_p\left(x\right)\rangle$. For this reason, all operations and relations on the n-ScIFN will still be considered as defined on the set:

$\mathit{n-ScIFN}{=}_{df}\{\langle \mu ,v\rangle \in [0,1]\times [0,1]:0\le S{c}_n\left(\mu \right)+S{c}_n\left(v\right)\le 1\}$.

Moreover, there is:

Fact 2.

$\mathit{n-ScIFN}=\left\{p\right(x):x\in X,p\in \mathit{n-ScIFS}\}$,

Any function $p:X\to \mathit{n-ScIFN}$ is the n-scaling intuitionistic fuzzy sets for sequences $\left\{Scn\right\},\left\{Sc{n}^{\prime }\right\}$ of scaling functions.

When $n=1$, then 1-ScIFSs are the intuitionistic fuzzy sets (IFS), which have been studied by Atanassow [25], and 2-ScIFSs for $S{c}_2\left(x\right)=S{c}_2^{\prime }\left(x\right)=x^2$ are the PFS of Yager [26].

From Definition 4 and from a simple arithmetic property: $0\le S{c}_{n+1}\left({\mu }_p\left(x\right)\right)+S{c}_{n+1}^{\prime }\left(v_p\left(x\right)\right)\le S{c}_n\left({\mu }_p\left(x\right)\right)+S{c}_n^{\prime }\left(v_p\left(x\right)\right)\le 1$ for $\langle {\mu }_p,v_p\rangle \in \mathit{n-ScIFS},$ there is:

Remark 2.

In order to simplify the description of the diagnosis granule model, it will be assumed in the further part of the paper that the equality holds for sequences of scaling functions: $\left\{Scn\right\}=\{Scn\prime \}$.

Theorem 5.

For any natural number $n>1$:

1. 

$\mathit{n-ScIFN}\subseteq (\mathit{n}+\mathit{1})\mathit{-ScIFN}\subseteq [0,1]\times [0,1],$

2. 

$\mathit{n-ScIFN}\subseteq (\mathit{n}+\mathit{1})\mathit{-ScIFN}.$

The membership and non-membership values $\langle \mu ,v\rangle \in [0,1]\times [0,1]$ such that $\mu +v>1$, are replaced with $\langle S{c}_n\left(\mu \right),S{c}_n\left(v\right)\rangle \in \mathit{1}\mathit{-ScIFN}$, for some n, when the power scale is entered. Then, the aggregation operations on the IFS can be extended to the n-ScIFN.

Theorem 6.

In any system, ${\mathbf{S}}_{\mathbf{Yager}}=\langle [0,1],{•}_t,{•}_s,{•}_p,{•}_l,0,1\rangle$, for any $\langle {\mu }_1,v_1\rangle ,\langle {\mu }_2,v_2\rangle \in \mathit{1}\mathit{-ScIFN}$ and numer $\lambda \in [0,1]$, the conditions are satisfied:

1. 

$\langle {\mu }_1{•}_s{\mu }_2,v_1{•}_t{v}_2\rangle \in \mathit{1}\mathit{-ScIFN}$,

2. 

$\langle {\mu }_1{•}_t{\mu }_2,v_1{•}_s{v}_2\rangle \in \mathit{1}\mathit{-ScIFN}$,

and in some systems of ${\mathbf{S}}_{\mathbf{Yager}}$, the below conditions are also satisfied:

3. 

$\langle \lambda {•}_l{\mu }_1,\lambda {•}_p{v}_1\rangle \in \mathit{1}\mathit{-ScIFN},$

4. 

$\langle \lambda {•}_p{\mu }_1,\lambda {•}_l{v}_1\rangle \in \mathit{1}\mathit{-ScIFN}.$

Proof. 

1.

$\langle {\mu }_1,v_1\rangle ,\langle {\mu }_2,v_2\rangle \in \mathit{1}\mathit{-ScIFN}$ iff ${\mu }_1+v_1\le 1,{\mu }_2+v_2\le 1$ iff ${\mu }_1\le 1-v_1,{\mu }_2\le 1-v_2$. Hence and from the monotonicity of the s-norm and its determination by the t-norms: ${\mu }_1{•}_s{\mu }_2\le (1-v_1){•}_s(1-v_2)=1-(1-(1-v_1){•}_s(1-v_2)=1-v_1{•}_t{v}_2$ iff ${\mu }_1{•}_s{\mu }_2+v_1{•}_t{v}_2\le 1$ iff $\langle {\mu }_1{•}_s{\mu }_2,v_1{•}_t{v}_2\rangle \in \mathit{1}\mathit{-ScIFN}$.

2.

$\langle {\mu }_1,v_1\rangle ,\langle {\mu }_2,v_2\rangle \in \mathit{1}\mathit{-ScIFN}$ iff $\langle v_1,{\mu }_1\rangle ,\langle v_2,{\mu }_2\rangle \in \mathit{1}\mathit{-ScIFN}$. Hence and from the point 1 it follows that: $\langle v_1{•}_s{v}_2,{\mu }_1{•}_t{\mu }_2\rangle \in \mathit{1}\mathit{-ScIFN}$. Which is equivalent to $\langle {\mu }_1{•}_t{\mu }_2,v_1{•}_s{v}_2\rangle \in \mathit{1}\mathit{-ScIFN}$.

3.

Let $\mu {•}_{s}v=max\{\mu ,v\},\mu {•}_{t}v=min\{\mu ,v\}$, then $f_s\left(x\right)=x,f_t\left(x\right)=1-x$ (see Example 1);

$\langle {\mu }_1,v_1\rangle \in \mathit{1}\mathit{-ScIFN}$ iff ${\mu }_1+v_1\le 1$;

$\lambda {•}_l{\mu }_1=f_s^{-1}\left[\lambda f_s\left({\mu }_1\right)\right]=\lambda {\mu }_1$ and $\lambda {•}_p{v}_1=f_s^{-1}\left[\lambda f_s\left(v_1\right)\right]=1-\lambda (1-v_1);$

$\lambda {•}_l{\mu }_1+\lambda {•}_p{v}_1=\lambda {\mu }_1+1-\lambda (1-v_1)=1-\lambda +\lambda ({\mu }_1+v_1).$

Since $\lambda ({\mu }_1+v_1)\le \lambda$, so $0\le 1-\lambda +\lambda ({\mu }_1+v_1)\le 1$.

4.

Let $\mu {•}_{s}v=max\{\mu ,v\},\mu {•}_{t}v=min\{\mu ,v\}$, then $f_s\left(x\right)=x,f_t\left(x\right)=1-x$;

$\langle {\mu }_1,v_1\rangle \in \mathit{1}\mathit{-ScIFN}$ iff ${\mu }_1+v_1\le 1$ iff $\langle v_1,{\mu }_1\rangle \in \mathit{1}\mathit{-ScIFN}$. Hence and from point 3, it follows that: $\langle \lambda {•}_l{v}_1,\lambda {•}_p{\mu }_1\rangle \in \mathit{1}\mathit{-ScIFN}$.

Which is equivalent to $\langle \lambda {•}_p{\mu }_1,\lambda {•}_l{v}_1\rangle \in \mathit{1}\mathit{-ScIFN}$.

   □

Remark 3.

Adopting the triangular norm generators from Example 2:

$\lambda {•}_l{\mu }_1={\lambda }^{1/p}{\mu }_1,\lambda {•}_p{v}_1=1-{\lambda }^{1/p}(1-v_1)$, for ${\mu }_1=1/2,v_1=7/8,\lambda =1/4$ and $p=2$ we get $\langle \lambda {•}_l{\mu }_1,\lambda {•}_p{v}_1\rangle =\langle 1/8,15/16\rangle \notin \mathit{1}\mathit{-ScIFN}$ (because $1<1/8+15/16$).

Theorem 7.

Allow in the system ${\mathbf{S}}_{\mathbf{Yager}}$, conditions 1–4 of Theorem 6 apply. Then for any natural number $n>1$ for any $\langle {\mu }_1,v_1\rangle ,\langle {\mu }_2,v_2\rangle \in \mathit{n-ScIFN}$ and number $\lambda \in [0,1]$, the following conditions are satisfied:

1. 

$\langle S{c}_n^{-1}\left(S{c}_n\left({\mu }_1\right){•}_{s}S{c}_n\left({\mu }_2\right)\right),S{c}_n^{-1}\left(S{c}_n\left(v_1\right){•}_{t}S{c}_n\left(v_2\right)\right)\rangle \in \mathit{n-ScIFN}$,

2. 

$\langle S{c}_n^{-1}\left(S{c}_n\left({\mu }_1\right){•}_{t}S{c}_n\left({\mu }_2\right)\right),S{c}_n^{-1}\left(S{c}_n\left(v_1\right){•}_{s}S{c}_n\left(v_2\right)\right)\rangle \in \mathit{n-ScIFN}$,

3. 

$\langle S{c}_n^{-1}\left(\lambda {•}_{l}S{c}_n\left({\mu }_1\right)\right),S{c}_n^{-1}\left(\lambda {•}_{p}S{c}_n\left(v_1\right)\right)\rangle \in \mathit{n-ScIFN}$,

4. 

$\langle S{c}_n^{-1}\left(\lambda {•}_{p}S{c}_n\left({\mu }_1\right)\right),S{c}_n^{-1}\left(\lambda {•}_{l}S{c}_n\left(v_1\right)\right)\rangle \in \mathit{n-ScIFN}$.

Proof. 

$\langle {\mu }_1,v_1\rangle ,\langle {\mu }_2,v_2\rangle \in \mathit{n-ScIFN}$ iff $\langle S{c}_n\left({\mu }_1\right),S{c}_n\left(v_1\right)\rangle ,\langle S{c}_n\left({\mu }_2\right),S{c}_n\left(v_2\right)\rangle \in \mathit{1}\mathit{-ScIFN}$.

Then the conditions of Theorem 6 equivalent to the above conditions 1–4 are satisfied.    □

Definition 6.

In the system ${\mathbf{S}}_{\mathbf{Yager}}$, the following aggregation operators, calledYager operators on n-ScIFN, are defined for any $\langle {\mu }_1,v_1\rangle ,\langle {\mu }_2,v_2\rangle \in \mathit{n-ScIFN}$ and the number $\lambda \in [0,1]$:

1. 

$\langle {\mu }_1,v_1\rangle \oplus \langle {\mu }_2,v_2\rangle =\langle S{c}_n^{-1}\left(S{c}_n\left({\mu }_1\right){•}_{s}S{c}_n\left({\mu }_2\right)\right),S{c}_n^{-1}\left(S{c}_n\left(v_1\right){•}_{t}S{c}_n\left(v_2\right)\right)\rangle ,$

2. 

$\langle {\mu }_1,v_1\rangle \otimes \langle {\mu }_2,v_2\rangle =\langle S{c}_n^{-1}\left(S{c}_n\left({\mu }_1\right){•}_{t}S{c}_n\left({\mu }_2\right)\right),S{c}_n^{-1}\left(S{c}_n\left(v_1\right){•}_{s}S{c}_n\left(v_2\right)\right)\rangle ,$

3. 

$\lambda \langle {\mu }_1,v_1\rangle =\langle S{c}_n^{-1}\left(\lambda {•}_{l}S{c}_n\left({\mu }_1\right)\right),S{c}_n^{-1}\left(\lambda {•}_{p}S{c}_n\left(v_1\right)\right)\rangle ,$

4. 

${\langle {\mu }_1,v_1\rangle }^{\lambda }=\langle S{c}_n^{-1}\left(\lambda {•}_{p}S{c}_n\left({\mu }_1\right)\right),S{c}_n^{-1}\left(\lambda {•}_{l}S{c}_n\left(v_1\right)\right)\rangle .$

When $\langle S{c}_n^{-1}\left(\lambda {•}_{l}S{c}_n\left({\mu }_1\right)\right),S{c}_n^{-1}\left(\lambda {•}_{p}S{c}_n\left(v_1\right)\right)\rangle \notin \mathit{n-ScIFN}$, then:

5. 

$\lambda \langle {\mu }_1,v_1\rangle =\langle 1,0\rangle ,$

or when $\langle S{c}_n^{-1}\left(\lambda {•}_{p}S{c}_n\left({\mu }_1\right)\right),S{c}_n^{-1}\left(\lambda {•}_{l}S{c}_n\left(v_1\right)\right)\rangle \notin \mathit{n-ScIFN}$, then:

6. 

${\langle {\mu }_1,v_1\rangle }^{\lambda }=\langle 0,1\rangle .$

Hence, in any system ${\mathbf{S}}_{\mathbf{Yager}}=\langle [0,1],{•}_t,{•}_s,{•}_p,{•}_l,0,1\rangle$, by Theorem 4, there is:

Theorem 8.

For any $\langle {\mu }_1,v_1\rangle ,\langle {\mu }_2,v_2\rangle ,\langle {\mu }_3,v_3\rangle \in \mathit{n-ScIFN}$ and any number $\lambda \in [0,1]$:

1. 

$\langle {\mu }_1,v_1\rangle \oplus \langle 0,1\rangle =\langle {\mu }_1,v_1\rangle ,\langle {\mu }_1,v_1\rangle \oplus \langle 1,0\rangle =\langle 1,0\rangle ,$

2. 

$\langle {\mu }_1,v_1\rangle \oplus \langle {\mu }_2,v_2\rangle =\langle {\mu }_2,v_2\rangle \oplus \langle {\mu }_1,v_1\rangle ,$

3. 

$(\langle {\mu }_1,v_1\rangle \oplus \langle {\mu }_2,v_2\rangle )\oplus \langle {\mu }_3,v_3\rangle =\langle {\mu }_1,v_1\rangle \oplus (\langle {\mu }_2,v_2\rangle \oplus \langle {\mu }_3,v_3\rangle ),$

4. 

$\langle {\mu }_1,v_1\rangle \otimes \langle 1,0\rangle =\langle {\mu }_1,v_1\rangle ,\langle {\mu }_1,v_1\rangle \oplus \langle 0,1\rangle =\langle 1,0\rangle ,$

5. 

$\langle {\mu }_1,v_1\rangle \otimes \langle {\mu }_2,v_2\rangle =\langle {\mu }_2,v_2\rangle \otimes \langle {\mu }_1,v_1\rangle ,$

6. 

$(\langle {\mu }_1,v_1\rangle \otimes \langle {\mu }_2,v_2\rangle )\otimes \langle {\mu }_3,v_3\rangle =\langle {\mu }_1,v_1\rangle \otimes (\langle {\mu }_2,v_2\rangle \otimes \langle {\mu }_3,v_3\rangle ),$

7. 

$\lambda (\langle {\mu }_1,v_1\rangle \oplus \langle {\mu }_2,v_2\rangle )=\lambda \langle {\mu }_1,v_1\rangle \oplus \lambda \langle {\mu }_2,v_2\rangle ,$

8. 

$(\langle {\mu }_1,v_1\rangle \otimes \langle {\mu }_2,v_2\rangle )\lambda =\langle {\mu }_1,v_1\rangle \lambda \otimes \langle {\mu }_2,v_2\rangle \lambda ,$

9. 

$({\lambda }_1+{\lambda }_2)\langle {\mu }_1,v_1\rangle ={\lambda }_1\langle {\mu }_1,v_1\rangle \oplus {\lambda }_2\langle {\mu }_1,v_1\rangle ,$

10. 

${\langle {\mu }_1,v_1\rangle }^{{\lambda }_1+{\lambda }_2}={\langle {\mu }_1,v_1\rangle }^{{\lambda }_1}\otimes {\langle {\mu }_1,v_1\rangle }^{{\lambda }_2}$.

4. Triangular Norms in n-ScIFN and n-ScIFN Algebra

Any set n-ScIFN can be ordered by the relation ${\le }_n$ defined as follows:

Definition 7.

For any $\langle {\mu }_1,v_1\rangle ,\langle {\mu }_2,v_2\rangle \in \mathit{n-ScIFN}$:

$$\langle {\mu }_1,v_1\rangle {\le }_n\langle {\mu }_2,v_2\rangle \mathit{iff}{\mu }_1\le {\mu }_2,v_1\ge v_2.$$

(6)

Fact 3.

For any $x,y\in [0,1]$,

1. 

$\langle 0,1\rangle {\le }_n\langle 0,x\rangle {\le }_n\langle 0,0\rangle {\le }_n\langle y,0\rangle {\le }_n\langle 1,0\rangle$,

2. 

$\langle 0,1\rangle {\le }_n\langle x,y\rangle {\le }_n\langle 1,0\rangle$, when $\langle x,y\rangle \in \mathit{n-ScIFN}$.

3. 

$\langle 0,1\rangle ={min}_n\mathit{n-ScIFN},\langle 1,0\rangle ={max}_n\mathit{n-ScIFN}.$

Analogous properties of the order relation ${\le }_n$ to the relation ≤ on real numbers enable the generalization of triangular norms into the set n-ScIFN.

Definition 8.

1. 

The operation ${\circ }_t:\mathit{n-ScIFN}\times \mathit{n-ScIFN}\to \mathit{n-ScIFN}$ is called thet-norm in the setn-ScIFNordered by the relation ${\le }_n$, when for any $\langle {\mu }_1,v_1\rangle ,\langle {\mu }_2,v_2\rangle ,\langle {\mu }_3,v_3\rangle \in \mathit{n-ScIFN}:$

(a) 

$\langle 0,1\rangle {\circ }_t\langle {\mu }_1,v_1\rangle =\langle 0,1\rangle ,\langle {\mu }_1,v_1\rangle {\circ }_t\langle 1,0\rangle =\langle {\mu }_1,v_1\rangle$—boundary conditions,

(b) 

$\langle {\mu }_1,v_1\rangle {\circ }_t\langle {\mu }_2,v_2\rangle {\le }_n\langle {\mu }_3,v_3\rangle {\circ }_t\langle {\mu }_2,v_2\rangle$, when $\langle {\mu }_1,v_1\rangle {\le }_n\langle {\mu }_3,v_3\rangle$—monotonicity,

(c) 

$\langle {\mu }_1,v_1\rangle {\circ }_t\langle {\mu }_2,v_2\rangle =\langle {\mu }_2,v_2\rangle {\circ }_t\langle {\mu }_1,v_1\rangle$—commutativity,

(d) 

$\langle {\mu }_1,v_1\rangle {\circ }_t\left(\langle {\mu }_2,v_2\rangle {\circ }_t\langle {\mu }_3,v_3\rangle \right)=\left(\langle {\mu }_1,v_1\rangle {\circ }_t\langle {\mu }_2,v_2\rangle \right){\circ }_t\langle {\mu }_3,v_3\rangle$—associativity.

2. 

The operation ${\circ }_s:\mathit{n-ScIFN}\times \mathit{n-ScIFN}\to \mathit{n-ScIFN}$ is called thes-norm in the setn-ScIFNordered by the relation ${\le }_n$, when for any $\langle {\mu }_1,v_1\rangle ,\langle {\mu }_2,v_2\rangle ,\langle {\mu }_3,v_3\rangle \in \mathit{n-ScIFN}$:

(a) 

$\langle 1,0\rangle {\circ }_s\langle {\mu }_1,v_1\rangle =\langle 1,0\rangle ,\langle {\mu }_1,v_1\rangle {\circ }_s\langle 0,1\rangle =\langle {\mu }_1,v_1\rangle$—boundary conditions,

(b) 

$\langle {\mu }_1,v_1\rangle {\circ }_s\langle {\mu }_2,v_2\rangle {\le }_n\langle {\mu }_3,v_3\rangle {\circ }_s\langle {\mu }_2,v_2\rangle$, when $\langle {\mu }_1,v_1\rangle {\le }_n\langle {\mu }_3,v_3\rangle$—monotonicity,

(c) 

$\langle {\mu }_1,v_1\rangle {\circ }_s\langle {\mu }_2,v_2\rangle =\langle {\mu }_2,v_2\rangle {\circ }_s\langle {\mu }_1,v_1\rangle$—commutativity,

(d) 

$\langle {\mu }_1,v_1\rangle {\circ }_s\left(\langle {\mu }_2,v_2\rangle {\circ }_s\langle {\mu }_3,v_3\rangle \right)=\left(\langle {\mu }_1,v_1\rangle {\circ }_s\langle {\mu }_2,v_2\rangle \right){\circ }_s\langle {\mu }_3,v_3\rangle$—associativity.

Theorem 9.

The Yager operator ⊗ on then-ScIFNis the t-norm in the setn-ScIFN. The Yager operator ⊕ in then-ScIFNis the s-norm.

Proof. 

Conditions (4)–(6) of Theorem 8 prove that the conditions (a), (c), (d) of Definition 7 (1) are satisfied with the t-norm in the set n-ScIFN by the operator ⊗. It is enough to prove that monotonicity is satisfied by this operation.

Let for any $\langle {\mu }_1,v_1\rangle ,\langle {\mu }_2,v_2\rangle ,\langle {\mu }_3,v_3\rangle \in \mathit{n-ScIFN}$:

• $\langle {\mu }_1,v_1\rangle \otimes \langle {\mu }_2,v_2\rangle =\langle S{c}_n^{-1}\left(S{c}_n\left({\mu }_1\right){•}_{t}S{c}_n\left({\mu }_2\right)\right),S{c}_n^{-1}\left(S{c}_n\left(v_1\right){•}_{s}S{c}_n\left(v_2\right)\right)\rangle ,$
• $\langle {\mu }_3,v_3\rangle \otimes \langle {\mu }_2,v_2\rangle =\langle S{c}_n^{-1}\left(S{c}_n\left({\mu }_3\right){•}_{t}S{c}_n\left({\mu }_2\right)\right),S{c}_n^{-1}\left(S{c}_n\left(v_3\right){•}_{s}S{c}_n\left(v_2\right)\right)\rangle ,$

and let $\langle {\mu }_1,v_1\rangle {\le }_n\langle {\mu }_3,v_3\rangle$. Then ${\mu }_1\le {\mu }_3,v_1\ge v_3$. Hence and form the monotonicity of t-norm and s-norm as well as the scale function $S{c}_n$, there are inequalities:

$S{c}_n\left({\mu }_1\right){•}_{t}S{c}_n\left({\mu }_2\right)\le S{c}_n\left({\mu }_3\right){•}_{t}S{c}_n\left({\mu }_2\right),$

$S{c}_n\left(v_3\right){•}_{s}S{c}_n\left(v_2\right)\le S{c}_n\left(v_1\right){•}_{s}S{c}_n\left(v_2\right))$ iff

$S{c}_n^{-1}\left(S{c}_n\left({\mu }_1\right){•}_{t}S{c}_n\left({\mu }_2\right)\right)\le S{c}_n^{-1}\left(S{c}_n\left({\mu }_3\right){•}_{t}S{c}_n\left({\mu }_2\right)\right),$

$S{c}_n^{-1}\left(S{c}_n\left(v_3\right){•}_{s}S{c}_n\left(v_2\right)\right)\ge S{c}_n^{-1}\left(S{c}_n\left(v_3\right){•}_{s}S{c}_n\left(v_2\right)\right)$ iff

$\langle S{c}_n^{-1}\left(S{c}_n\left({\mu }_1\right){•}_{t}S{c}_n\left({\mu }_2\right)\right),S{c}_n^{-1}\left(S{c}_n\left(v_1\right){•}_{s}S{c}_n\left(v_2\right)\right)\rangle {\le }_n$

$\langle S{c}_n^{-1}\left(S{c}_n\left({\mu }_3\right){•}_{t}S{c}_n\left({\mu }_2\right)\right),S{c}_n^{-1}\left(S{c}_n\left(v_3\right){•}_{s}S{c}_n\left(v_2\right)\right)\rangle$ iff $\langle {\mu }_1,v_1\rangle \otimes \langle {\mu }_2,v_2\rangle {\le }_n\langle {\mu }_3,v_3\rangle \otimes \langle {\mu }_2,v_2\rangle .$

The fulfillment by the operator ⊕ of the conditions of Definition 7 (2) of the s-norm in the n-ScIFN set is proved analogously.    □

The monotonicity of the operators $\oplus ,\otimes$ describes important Theorem:

Theorem 10.

For any $\langle {\mu }_1,v_1\rangle ,\langle {\mu }_2,v_2\rangle ,\langle {\mu }_3,v_3\rangle \in \mathit{n-ScIFN}$:

1. 

$\langle {\mu }_1,v_1\rangle \otimes \langle {\mu }_2,v_2\rangle {\le }_n\langle {\mu }_1,v_1\rangle$ and $\langle {\mu }_1,v_1\rangle \otimes \langle {\mu }_2,v_2\rangle {\le }_n\langle {\mu }_2,v_2\rangle ,$

2. 

$\langle {\mu }_1,v_1\rangle {\le }_n\langle {\mu }_1,v_1\rangle \oplus \langle {\mu }_2,v_2\rangle$ and $\langle {\mu }_2,v_2\rangle {\le }_n\langle {\mu }_1,v_1\rangle \oplus \langle {\mu }_2,v_2\rangle ,$

3. 

$\langle {\mu }_1,v_1\rangle \otimes \langle {\mu }_2,v_2\rangle {\le }_n\langle {\mu }_1,v_1\rangle \oplus \langle {\mu }_2,v_2\rangle ,$

4. 

if $\langle {\mu }_1,v_1\rangle {\le }_n\langle {\mu }_3,v_3\rangle$, then $\langle {\mu }_1,v_1\rangle \otimes \langle {\mu }_2,v_2\rangle {\le }_n\langle {\mu }_3,v_3\rangle \otimes \langle {\mu }_2,v_2\rangle$ and $\langle {\mu }_1,v_1\rangle \oplus \langle {\mu }_2,v_2\rangle {\le }_n\langle {\mu }_3,v_3\rangle \oplus \langle {\mu }_2,v_2\rangle ,$

5. 

if $\langle {\mu }_1,v_1\rangle {\le }_n\langle {\mu }_2,v_2\rangle$, then for any number $\lambda \in [0,1],\lambda \langle {\mu }_1,v_1\rangle \otimes {\le }_n\lambda \langle {\mu }_2,v_2\rangle$ and ${\langle {\mu }_1,v_1\rangle }^{\lambda }\otimes {\le }_n{\langle {\mu }_2,v_2\rangle }^{\lambda }.$

Proof. 

From fact 3 (2) it follows $\langle {\mu }_1,v_1\rangle {\le }_n\langle 1,0\rangle ,\langle {\mu }_2,v_2\rangle {\le }_n\langle 1,0\rangle$. Hence, from monotonicity of ⊗, by Definition 8 (1), there is a proof of the point (1):

$\langle {\mu }_1,v_1\rangle \otimes \langle {\mu }_2,v_2\rangle {\le }_n\langle 1,0\rangle \otimes \langle {\mu }_2,v_2\rangle =\langle {\mu }_2,v_2\rangle$,

$\langle {\mu }_1,v_1\rangle \otimes \langle {\mu }_2,v_2\rangle {\le }_n\langle {\mu }_1,v_1\rangle \otimes \langle 1,0\rangle =\langle {\mu }_1,v_1\rangle .$

From fact 3 (2) it follows $\langle 0,1\rangle {\le }_n\langle {\mu }_1,v_1\rangle ,\langle 0,1\rangle {\le }_n\langle {\mu }_2,v_2\rangle$. Hence, from monotonicity of ⊕, by Definition 8 (1), there is a proof of point (2):

$\langle {\mu }_1,v_1\rangle =\langle 0,1\rangle \otimes \langle {\mu }_1,v_1\rangle {\le }_n\langle {\mu }_1,v_1\rangle \otimes \langle {\mu }_2,v_2\rangle ,$

$\langle {\mu }_2,v_2\rangle =\langle 0,1\rangle \otimes \langle {\mu }_2,v_2\rangle {\le }_n\langle {\mu }_1,v_1\rangle \otimes \langle {\mu }_2,v_2\rangle .$

Point (3) of Theorem follows the transitivity of the relations ${\le }_n$, point (4) is obtained based on Theorem 9 and point (5) follows the monotonicity of linear and power repetition operations.    □

For relation ${\le }_n$ and any $A\subseteq \mathit{n-ScIFN}$, it is not always possible to define supremum and infinitum operations in terms of mathematical logic. Therefore it is accepted:

Definition 9.

For any $A\subseteq \mathit{n-ScIFN}$ and relations ${\le }_n$ then-supremumA is ${sup}_{n}A$:

• For $A=⌀,{sup}_{n}A=\langle 0,1\rangle ,$
• For $A\ne ⌀$: ${sup}_{n}A{=}_{df}\langle sup\{u:$ exists $v\in [0,1]$ such that $\langle u,v\rangle \in A\},inf\{v:$ exists $u\in [0,1]$ such that $\langle u,v\rangle \in A\}\rangle$

and then-infimumA is ${inf}_{n}A$:

• For $A=⌀,{inf}_{n}A=\langle 0,1\rangle ,$
• For $A\ne ⌀$: ${inf}_{n}A{=}_{df}\langle inf\{u:$ exists $v\in [0,1]$ such that $\langle u,v\rangle \in A\}$,$sup\{v:$ exists $u\in [0,1]$ such that $\langle u,v\rangle \in A\}\rangle .$

From Definitions 7 and 9 there is:

Fact 4.

For any $A\subseteq \mathit{n-ScIFN}$ and $\langle u,v\rangle \in A\}$, ${inf}_{n}A{\le }_n\langle u,v\rangle {\le }_{n}su{p}_{n}A$.

5. Diagnostic Information Granules

The concept of the information granule system presented in [20] has models in the theories of: sets, probability, possible data sets in evidence systems, fuzzy sets, rough sets, and shadowed sets [20,42,43]. Here this concept is expanded and a broader model of information granules is formulated. It is called the system of diagnostic information granules.

The diagnosis of the object $x\in X$ is a pair of fuzzy data symptoms $u\in [0,1]$ and faults $v\in [0,1]$ of the behavior of the diagnosed object in the diagnostic process. A diagnosis instance is understood as a triple $\langle x,\langle u,v\rangle \rangle$, such that if symptoms $u=0$, then faults $v\ne 1$ (if there is no symptom, the error cannot be certain). Let U denote the set of diagnosis instances. The symptom-faults two-dimensional granule system was extended to include operations of the degree of repetition of diagnostic procedures: linear and power, for parallel and serial measurements, respectively. In order to determine the diagnostic information granules (shortly: diagnostic granules or granules), a first distinction is made between the set of the granule instances $G_{inst}$ and the set of the granule of instances sets $G_{set}$.

5.1. The Diagnostic Granule System: Its Language and Interpretation

The set granules is defined by the set of instance granules. In other words, elements of the instance set granules are instance granules. In summary, the knowledge about operations and relations in n-ScIFN introduced above can, for the fuzzy sets of n-ScIFS in the X space of the information instance, be defined as a diagnostic information granule system as follows:

Definition 10.

Relational structure:

$${\mathit{G}}_{\mathit{diag}}=\langle G,{\cup }_G,{\cap }_G,{\setminus }_G,{}^{\prime G},{\left\{\right\}}_G,{\left\{{{\displaystyle \sum }}_G^{\lambda }\right\}}_{\lambda \in [0,1]},{\left\{{{\displaystyle \prod }}_G^{\lambda }\right\}}_{\lambda \in [0,1]},{\in }_G,{\subseteq }_G,{=}_G,0_G,1_G,G_{inst},G_{set},G_0\rangle ,$$

(7)

will be called thesystem of diagnostic granules, where:

• Theset of diagnostic granules$G=G_{inst}\cup G_{set}$ is the sum of the sets: $G_{inst}$—instance granulesand $G_{set}$—set granules,
• Operations ${\cup }_G,{\cap }_G$ are thesumandproductof two granules, as well as the generalized sum and product on the granule sets,
• ${\setminus }_G$ is an operation ofgranule difference,
• ${\prime }^G$ is an operation ofgranule closure,
• ${\left\{\right\}}_G$ is an operation ofsingleton,
• ${{\displaystyle \sum }}_G^{\lambda }$ is the operation of thelinear repetition of granuleswith the degree λ,
• ${{\displaystyle \prod }}_G^{\lambda }$ is the operation of thepower repetition of granuleswith the degree λ,
• $0_G$ is anempty granulefor an empty set and $1_G$ is afull granulefor a set G,
• $G_0$ is the set of distinguished granules calledinstance singleton granules.

For granules of instance sets, the following relations are defined:

• ${\in }_G$ is the relation ofbeing an element of the granule,
• ${\subseteq }_G$ is thegranule inclusion relation,
• ${=}_G$ is thegranule closeness relation.

The symbols of the relational structure ${\mathit{G}}_{\mathit{diag}}$ and the language expressions of the diagnostic granule system are interpreted as follows:

1.

$G_{set}=\mathit{n-ScIFN}$,

2.

$G_{inst}=\{\langle a,\langle {\mu }_p\left(a\right),v_p\left(a\right)\rangle \rangle :p\in G_{set}$, $a\in X,$$\langle {\mu }_p\left(a\right),v_p\left(a\right)\rangle \ne \langle 0,1\rangle$ and for $x\in X,x\ne a,\langle {\mu }_p\left(x\right),v_p\left(x\right)\rangle =\langle 0,1\rangle \}$,

3.

$G_0=\{p\in G_{set}:$ exists $a\in X,\langle {\mu }_p\left(a\right),v_p\left(a\right)\rangle \ne \langle 0,1\rangle$ and for $x\in X,x\ne a,\langle {\mu }_p\left(x\right),v_p\left(x\right)\rangle =\langle 0,1\rangle \}$.

For any $A\subseteq G_{set}$:

4.

${\cup }_{G}A=\{\langle {\mu }_p\left(x\right),v_p\left(x\right)\rangle :\langle {\mu }_p\left(x\right),v_p\left(x\right)\rangle ={sup}_n\{\langle {\mu }_e\left(x\right),v_e\left(x\right)\rangle :e\in A\}$ and $x\in X\}$,

5.

The operation ${\left\{\right\}}_G:G_{inst}\to G_0$, determining a singleton instance, is given by the formula: if for $p\in G_0$ and for $a\in X,\langle a,\langle {\mu }_p\left(a\right),v_p\left(a\right)\rangle \rangle \in G_{inst}$, then ${\left\{\langle a,\langle {\mu }_p\left(a\right),v_p\left(a\right)\rangle \rangle \right\}}_G=p$.

$G_0$ is a set of granules called instance singleton granules such that a function is given ${\left\{\right\}}_G:G_{inst}\to G_0$ and $G_0=\{x\in G_{set}:\exists z\in G_{inst}(x={\left\{z\right\}}_G)\}$.

For any $p_1,p_2\in G_{set}$ and any number $\lambda \in [0,1]$:

6. 

$p_1{\cup }_G{p}_2=\{\langle x,\langle {\mu }_{p1}\left(x\right),v_{p1}\left(x\right)\rangle \oplus \langle {\mu }_{p2}\left(x\right),v_{p2}\left(x\right)\rangle \rangle :x\in X\}$,

7. 

$p_1{\cap }_G{p}_2=\{\langle x,\langle {\mu }_{p1}\left(x\right),v_{p1}\left(x\right)\rangle \otimes \langle {\mu }_{p2}\left(x\right),v_{p2}\left(x\right)\rangle \rangle :x\in X\}$,

8. 

${{\displaystyle \sum }}_G^{\lambda }{p}_1=\{\langle x,\lambda \langle {\mu }_{p1}\left(x\right),v_{p1}\left(x\right)\rangle \rangle :x\in X\}$,

9. 

${{\displaystyle \prod }}_G^{\lambda }{p}_1=\{\langle x,{\langle {\mu }_{p1}\left(x\right),v_{p1}\left(x\right)\rangle }^{\lambda }\rangle :x\in X\}$,

10. 

$p_1{\subseteq }_G{p}_2$ iff for any $x\in X,\langle {\mu }_{p1}\left(x\right),v_{p1}\left(x\right)\rangle {\le }_n\langle {\mu }_{p2}\left(x\right),v_{p2}\left(x\right)\rangle$, i.e., ${\mu }_{p1}\left(x\right)\le {\mu }_{p2}\left(x\right)$, and  $v_{p1}\left(x\right)\ge v_{p2}\left(x\right)$,

11. 

$p_1{=}_G{p}_2$ iff ${\cup }_G\{z\in G_0:\left(z{\subseteq }_G{p}_1\right)\iff \left(z{\subseteq }_G{p}_2\right)\}=1_G$.

For any $e\in G_{inst},p\in G_{set}$:

12. 

$e{\in }_{G}p$ iff ${\left\{e\right\}}_G{\subseteq }_{G}p$.

Symbols $0_G,1_G$, are interpreted by formulas:

13. 

$p=0_G$ iff for any $x\in X,\langle {\mu }_p\left(x\right),v_p\left(x\right)\rangle =\langle 0,1\rangle ,$

14. 

$p=1_G$ iff for any $x\in X,\langle {\mu }_p\left(x\right),v_p\left(x\right)\rangle =\langle 1,0\rangle .$

Similarly to the set theory, it can be assumed that:

15. 

$y{\prime }^G={\cup }_G\{t\in G_0:\neg t{\subseteq }_{G}y\}$,

16. 

$y{\setminus }_{G}z={\cup }_G\{t\in G_0:t{\subseteq }_{G}y\wedge \neg t{\subseteq }_{G}z)\}$.

By applying the above conditions (1)–(16), and simple calculus, Definition 9, Fact 5, and Theorem 11 are proved.

Fact 5.

1. 

${\cup }_G⌀=0_G$,

2. 

${\cup }_G{G}_0=1_G$,

3. 

$y{\prime }^G=1_G$,

4. 

$y{\setminus }_G{1}_G=0_G$,

5. 

$y{\setminus }_{G}z{\subseteq }_{G}y{\cap }_{G}z{\prime }^G=y$.

Theorem 11.

For any $p_1,p_2\in G_{set}$ and $A,B\subseteq G_{set}$:

1. 

$p_1{\subseteq }_G{p}_2$ iff for any $t\in G_0$, if $t{\subseteq }_G{p}_1$, then $t{\subseteq }_G{p}_2$,

2. 

$p_1{\subseteq }_G{p}_2$ iff for any $e\in G_{inst}$, if $x{\in }_G{p}_1$, then $e{\in }_G{p}_2$,

3. 

$p_1{=}_G{p}_2$ iff $p_1{\subseteq }_G{p}_2$ and $p_2{\subseteq }_G{p}_1$,

4. 

$p_1{=}_G{p}_2$ iff $p_1=p_2$,

5. 

$p_1={\cup }_G\{t\in G_0:t{\subseteq }_G{p}_1\}$,

6. 

if $A\subseteq B$, then ${\cup }_{G}A{\subseteq }_G{\cup }_{G}B$,

7. 

for any $e\in G_0$ exists $p\in A$ such that $e{\subseteq }_{G}p$ iff $e{\subseteq }_G{\cup }_{G}A$.

Proof. 

For example, condition (5) of Theorem 11 is proven.

Let $e\in G_0$. If $e{\subseteq }_G{p}_1$, then $e{\subseteq }_G{\cup }_G\{t\in G_0:t{\subseteq }_G{p}_1\}$.

If $e{\subseteq }_{G}p={\cup }_G\{t\in G_0:t{\subseteq }_G{p}_1\}$, then for any $x\in X,e\left(x\right)=\langle u_e\left(x\right),v_e\left(x\right)\rangle {\le }_n\langle {\mu }_p\left(x\right),v_p\left(x\right)\rangle ={sup}_n\{\langle {\mu }_t\left(x\right),v_t\left(x\right)\rangle :t\in G_0,t{\subseteq }_G{p}_1\}$.

Since $\langle u_e\left(a\right),v_e\left(a\right)\rangle {\le }_n\langle {\mu }_p\left(x\right),v_p\left(x\right)\rangle {\le }_n\langle {\mu }_{p1}\left(x\right),v_{p1}\left(x\right)\rangle$, so $e{\subseteq }_G{p}_1$.

Hence, based on condition (3) of Theorem 11, there is $p_1={\cup }_G\{t\in G_0:t{\subseteq }_G{p}_1\}$.    □

Hence and based on Theorem 8, introducing a notation agreement: ${{\displaystyle \sum }}_G^{\lambda }p{=}_{df}\lambda p,{{\displaystyle \prod }}_G^{\lambda }p{=}_{df}{p}^{\lambda }$, we get:

Theorem 12.

For any $p_1,p_2,p_3\in \mathit{n-ScIFN}$ and any number $\lambda ,{\lambda }_1,{\lambda }_2\in [0,1]$:

1. 

$p_1{\cup }_G{0}_G=p_1,p_1{\cup }_G{1}_G=1_G$,

2. 

$p_1{\cup }_G{p}_2=p_2{\cup }_G{p}_1,$

3. 

$\left(p_1{\cup }_G{p}_2\right){\cup }_G{p}_3=p_1{\cup }_G\left(p_2{\cup }_G{p}_3\right)$,

4. 

$p_1{\cap }_G{1}_G=p_1,p_1{\cap }_G{0}_G=0_G$,

5. 

$p_1{\cap }_G{p}_2=p_2{\cap }_G{p}_1,$

6. 

$\left(p_1{\cap }_G{p}_2\right){\cap }_G{p}_3=p_1{\cap }_G\left(p_2{\cap }_G{p}_3\right)$,

7. 

$\lambda \left(p_1{\cup }_G{p}_2\right)=\lambda p_1{\cup }_G\lambda p_2,$

8. 

${\left(p_1{\cap }_G{p}_2\right)}^{\lambda }=p_1^{\lambda }{\cap }_G{p}_2^{\lambda }$,

9. 

$({\lambda }_1+{\lambda }_2)p_1={\lambda }_1{p}_1{\cup }_G{\lambda }_2{p}_1$,

10. 

$p_1^{\lambda 1+\lambda 2}=p_1^{\lambda 1}{\cap }_G{p}_1^{\lambda 2}$.

Definition 11.

Conditions (1)–(10) in Theorem 12 define thegranular algebra of diagnosisdefined on the n-ScIFS set.

5.2. Information Granule System Database

Using the constructions on rough sets given by E. Bryniarski in the papers [42,44], it is possible to define the simplest sub-system of the granules system, in which from the set of instances $G_{inst}$ using the operation ${\left\{\right\}}_G$, transforming the instances as elementary granules (granular elements) into granular singletons and operation of singleton summation ${\cup }_G$, any information granule is obtained.

Definition 12.

Any granule system $\mathit{G}$:

$${\mathit{G}}_{\mathit{base}}=\langle G,{\left\{\right\}}_G,{\cup }_G,{\subseteq }_G,{=}_G,0_G,1_G,G_{inst},G_{set},G_0\rangle ,$$

(8)

is called theinformation granule database, when the elements of the G set are called granules, operations ${\left\{\right\}}_G$ is the singleton, ${\cup }_G$ is the granule sum, ${\subseteq }_G$ is the inclusion relation, ${=}_G$ is the closeness relation, $0_G$ is the empty granule and $1_G$ is the full granule, and $G_{inst}$ is the set of instance granules, $G_{set}$ is the set of set granules, and $G_0$ is the singleton set of instance granules. The system ${\mathit{G}}_{\mathit{base}}$ satisfies the following axioms:

1. 

$G_0\subseteq G_{set}$,

2. 

$G=G_{inst}\cup G_{set}$,

3. 

$G_{inst}\cap G_{set}=⌀$.

For any $x\in G_{set}$:

4. 

$x{\subseteq }_{G}x$,

5. 

$0_G{\subseteq }_{G}x$,

6. 

If $x\ne 0_G$, then it is not true that $x{\subseteq }_G{0}_G$.

There is some function ${\left\{\right\}}_G:G_{inst}\to G_0$, such that:

7. 

For any $x\in G_{inst},{\left\{x\right\}}_G{\subseteq }_G{1}_G$,

8. 

$G_0=\{x\in G_0:\exists z\in G_{inst}(x={\left\{z\right\}}_G)\}$.

There is some function ${\cup }_G:\phi \left(G_0\right)\to G_{set}$, such that:

9. 

${\cup }_G⌀=0_G,$

10. 

${\cup }_G{G}_0=1_G$,

11. 

for any $y\in G_{set},y={\cup }_G\{x\in G_0:x{\subseteq }_{G}y\}$,

12. 

for any $x,y\in G_{set},\left(x{=}_{G}y\right){\iff }_{df}{\cup }_G\{z\in G_0:\left(z{\subseteq }_{G}x\right)\iff \left(z{\subseteq }_{G}y\right)\}=1_G$.

The information granule database ${\mathit{G}}_{\mathit{base}}$ is a diagnostic granule subsystem. When we join to the conditions (1)–(12), a Definition: for any $x\in G_{inst}$ and $y\in G_{set}$, $x{\in }_{G}y$ iff ${\left\{x\right\}}_G{\subseteq }_{G}y$.

Then we get the following Theorem.

Theorem 13.

1. 

$\{x\in G_{inst}:x{\in }_G{0}_G\}=⌀,$

2. 

$\{x\in G_{inst}:\exists z\in G_0\left(x{\in }_{G}z\right)\}=G_{inst}$,

3. 

$\{z\in G_{set}:\exists x\in G_{inst}\left(x{\in }_{G}z\right)\}=G_{set}$,

4. 

$y={\cup }_G\{{\left\{x\right\}}_G:x{\in }_{G}y\}$, for any $y\in G_{set}$.

Theorem 7 describes the set granules analogously to the classical set theory. The following theorem answers question about the analogy of operation on diagnostic granules to set operations.

Theorem 14.

In the granule database ${\mathit{G}}_{\mathit{diag}}$, the diagnostic granules $y{\prime }^G,y{\cap }_{G}z,y{\cup }_{G}z,y{\setminus }_{G}z$, are described by formulas:

1. 

$y{\prime }^G={\cup }_G\{z\in G_0:\exists x\in G_{inst}(x{\in }_{G}z\wedge \neg x{\in }_{G}y)\}$,

2. 

$y{\cap }_{G}z{\subseteq }_G{\cup }_G\{t\in G_0:\exists x\in G_{inst}(x{\in }_{G}t\wedge x{\in }_{G}y\wedge x{\in }_{G}z)\}$,

3. 

${\cup }_G\{t\in G_0:\exists x\in G_{inst}(x{\in }_{G}t\wedge (x{\in }_{G}y\vee x{\in }_{G}z))\}{\subseteq }_{G}y{\cup }_{G}z,$

4. 

$y{\setminus }_{G}z={\cup }_G\{t\in G_0:\exists x\in G_{inst}(x{\in }_{G}t\wedge x{\in }_{G}y\wedge \neg x{\in }_{G}z)\}$.

Proof. 

Conditions (1) and (4) are fulfilled by axioms (15) and (16) of Definition 10 of the diagnostic granule system and of Fact 5. Conditions (2) and (3) are proofed based on Theorems 10 and 11 and Definition 10.

Proof of condition (2).

$y{\cap }_{G}z{\subseteq }_{G}y$ and $y{\cap }_{G}z{\subseteq }_{G}z.$

Hence $\{t\in G_0:t{\subseteq }_{G}y{\cap }_{G}z\}\subseteq \{t\in G_0:t{\subseteq }_{G}y\wedge t{\subseteq }_{G}z)\}=\{t\in G_0:\exists x\in G_{inst}(x{\in }_{G}t\wedge x{\in }_{G}y\wedge x{\in }_{G}z)\}$.

Moreover, $y{\cap }_{G}z{\subseteq }_G{\cup }_G\{t\in G_0:t{\subseteq }_{G}y{\cap }_{G}z\}{\subseteq }_G{\cup }_G\{t\in G_0:t{\subseteq }_{G}y\wedge t{\subseteq }_{G}z\}={\cup }_G\{t\in G_0:\exists x\in G_{inst}(x{\in }_{G}t\wedge x{\in }_{G}y\wedge x{\in }_{G}z)\}.$

Proof of condition (3).

$y{\subseteq }_{G}y{\cup }_{G}z$ and $z{\subseteq }_{G}y{\cup }_{G}z$.

Hence $\{t\in G_0:\exists x\in G_{inst}(x{\in }_{G}t\wedge (x{\in }_{G}y\vee x{\in }_{G}z))\}=\{t\in G_0:t{\subseteq }_{G}y\vee t{\subseteq }_{G}z\}\subseteq \{t\in G_0:t{\subseteq }_{G}y{\cup }_{G}z\}$.

Moreover ${\cup }_G\{t\in G_0:\exists x\in G_{inst}(x{\in }_{G}t\wedge (x{\in }_{G}y\vee x{\in }_{G}z))\}{\subseteq }_G{\cup }_G\{t\in G_0:t{\subseteq }_{G}y{\cup }_{G}z\}=y{\cup }_{G}z.$    □

6. An Example of a Standard Diagnostic Granule System

6.1. Standard Diagnostic Granule System

For any $\langle {\mu }_1,v_1,\rangle ,\langle {\mu }_2,v_2\rangle \in \mathit{n-ScIFS},\mu \in [0,1]$ there are the following formulas:

S1.${\mu }_1{•}_s{v}_1=max\{{\mu }_1,v_1\},$

${\mu }_1{•}_t{v}_1=min\{{\mu }_1,v_1\}$,

and then $f_s\left(x\right)=x,f_t\left(x\right)=1-x$;

$\lambda {•}_l\mu =f_s^{-1}\left[\lambda f_s\left(\mu \right)\right]=\lambda \mu$ and

$\lambda {•}_p\mu =f_t^{-1}\left[\lambda f_t\mu \right)]=1-\lambda (1-\mu )$;

S2.$\langle {\mu }_1,v_1\rangle \oplus \langle {\mu }_2,v_2\rangle =$

$\langle S{c}_n^{-1}\left(S{c}_n\left({\mu }_1\right){•}_{s}S{c}_n\left({\mu }_2\right)\right),S{c}_n^{-1}\left(S{c}_n\left(v_1\right){•}_{t}S{c}_n\left(v_2\right)\right)\rangle =$

$\langle max\{u_1,u_2\},min\{v_1,v_2\}\rangle ,$

S3.$\langle {\mu }_1,v_1\rangle \otimes \langle {\mu }_2,v_2\rangle =$

$\langle S{c}_n^{-1}\left(S{c}_n\left({\mu }_1\right){•}_{t}S{c}_n\left({\mu }_2\right)\right),S{c}_n^{-1}\left(S{c}_n\left(v_1\right){•}_{s}S{c}_n\left(v_2\right)\right)\rangle =$

$\langle min\{u_1,u_2\},max\{v_1,v_2\}\rangle ,$

S4.$\lambda \langle {\mu }_1,v_1\rangle =$

$\langle S{c}_n^{-1}\left(\lambda {•}_{l}S{c}_n\left({\mu }_1\right)\right),S{c}_n^{-1}\left(\lambda {•}_{p}S{c}_n\left(v_1\right)\right)\rangle =$

$\langle S{c}_n^{-1}\left(\lambda S{c}_n\left({\mu }_1\right)\right),S{c}_n^{-1}(1-\lambda (1-S{c}_n\left(v_1\right)))\rangle ,$

S5.$\langle {\mu }_1,v_1\rangle \lambda =$

$\langle S{c}_n^{-1}\left(\lambda {•}_{p}S{c}_n\left({\mu }_1\right)\right),S{c}_n^{-1}\left(\lambda {•}_{l}S{c}_n\left(v_1\right)\right)\rangle =$

$\langle S{c}_n^{-1}(1-\lambda (1-S{c}_n\left(u_1\right))),S{c}_n^{-1}\left(\lambda S{c}_n\left(v_1\right)\right)\rangle .$

It is assumed that:

• Symptom-fault diagnosis $(s,f)$ corresponds to the diagnostic granule $p=({\mu }_p,v_p)\in \mathit{n-ScIFS}$ defined by fuzzy sets ${\mu }_p:X\to [0,1],v_p:X\to [0,1]$ and the values of the function ${\mu }_i,v_j$ in the scale $S{c}_n$ satisfy the condition for any $x\in X$:

  $$0\le S{c}_n\left({\mu }_p\left(x\right)\right)+S{c}_n\left(v_p\left(x\right)\right)\le 1.$$

  (9)
• For any diagnostic granule $p=({\mu }_p,v_p)$, granules $\lambda p$ is a result of parallel repetition of an alternative of the diagnosis p, and $p^{\lambda }$ is the result of linear repetition of the conjunction of diagnosis p. This is due to the inaccuracy, uncertainty, or imprecision of the diagnosis results. The degree of diagnosis that requires repetitions is the number $\lambda \in (0,1]$. If the diagnosis does not need to be repeated and is certain, then $\lambda =1$. The diagnosis can be repeated multiple times until it is certain. For example, if we want to repeat diagnosis maximally 5 times, then $\lambda =1-i/5$, where i is the number of repetitions, then $\lambda =1,4/5,3/5,2/5,1/5,0$.

Definition 13.

The diagnostic granule system determined by the assumptions described above will be called thestandard diagnostic granule system.

Theorem 15.

In the standard diagnostic granules system with the scaling function $S{c}_n$, for any granules $y,z\in \mathit{n-ScIFS}$, the following conditions apply:

1. 

$y{\cap }_{G}z{=}_{df}{\cup }_G\{t\in G_0:\exists x\in G_{inst}(x{\in }_{G}t\wedge x{\in }_{G}y\wedge x{\in }_{G}z)\},$

2. 

$y{\cup }_{G}z{=}_{df}{\cup }_G\{y,z\}={\cup }_G\{t\in G_0:\exists x\in G_{inst}(x{\in }_{G}t\wedge (x{\in }_{G}y\vee x{\in }_{G}z))\}.$

Proof. 

The proof of the point (1):

$x=\langle a,\langle {\mu }_p\left(a\right),v_{p1}\left(a\right)\rangle \rangle {\in }_G{\cup }_G\{t\in G_0:\exists x\in G_{inst}(x{\in }_{G}t\wedge x{\in }_{G}y\wedge x{\in }_{G}z)\}$ iff $\langle {\mu }_p\left(a\right),v_p\left(a\right)\rangle {\le }_n\langle {\mu }_y\left(a\right),v_y\left(a\right)\rangle \wedge \langle {\mu }_p\left(a\right),v_p\left(a\right)\rangle {\le }_n\langle {\mu }_z\left(a\right),v_z\left(a\right)\rangle$ iff ${\mu }_p\left(a\right)\le {\mu }_y\left(a\right)\wedge v_y\left(a\right)\le v_p\left(a\right)\wedge {\mu }_p\left(a\right)\le {\mu }_z\left(a\right)\wedge v_z\left(a\right)\le v_p\left(a\right)$.

Hence ${\mu }_p\left(a\right)\le inf\{{\mu }_y\left(a\right),{\mu }_z\left(a\right)\}\wedge max\{v_y\left(a\right),v_z\left(a\right)\}\le v_p\left(a\right)$ iff $\langle {\mu }_p\left(a\right),v_p\left(a\right)\rangle {\le }_n\langle inf\{{\mu }_y\left(a\right),{\mu }_z\left(a\right)\},max\{v_y\left(a\right),v_z\left(a\right)\}\rangle =\langle {\mu }_y\left(a\right),v_y\left(a\right)\rangle \otimes \langle {\mu }_z\left(a\right),v_z\left(a\right)\rangle$ iff $x=\langle a,\langle {\mu }_p\left(a\right),v_{p1}\left(a\right)\rangle \rangle {\in }_{G}y{\cap }_{G}z.$

Therefore ${\cup }_G\{t\in G_0:\exists x\in G_{inst}(x{\in }_{G}t\wedge x{\in }_{G}y\wedge x{\in }_{G}z)\}{\subseteq }_{G}y{\cap }_{G}z$.

From Theorem 14: $y{\cap }_{G}z{\subseteq }_G{\cup }_G\{t\in G_0:\exists x\in G_{inst}(x{\in }_{G}t\wedge x{\in }_{G}y\wedge x{\in }_{G}z)\}.$ Hence and from Theorem 11: $y{\cap }_{G}z={\cup }_G\{t\in G_0:\exists x\in G_{inst}(x{\in }_{G}t\wedge x{\in }_{G}y\wedge x{\in }_{G}z)\}$.

The proof of the point (2) is analogous.    □

6.2. The First Element of the Sequence Sc_n That Satisfies the n-ScIFS

Table 1. Symptoms, faults, fuzzy degree, and determination of the number n for n-ScIFS.

Diagnosis $\mathit{p}$	Object $\mathit{o}$	Symptom $\mathit{s}$	Fault $\mathit{f}$	Number n for ${\mathit{Sc}}_{\mathit{n}}\left(\mathit{x}\right)=\mathit{x}/\mathit{n}$	Number n for ${\mathit{Sc}}_{\mathit{n}}\left(\mathit{x}\right)=\mathit{1}-{\mathbf{log}}_{(\mathit{1}-\mathit{1}/\mathit{n})}\mathit{x}$	Number n for ${\mathit{Sc}}_{\mathit{n}}\left(\mathit{x}\right)={\mathit{x}}^{\mathit{n}}$
	1	0.5	0.4	1	1	1
	2	0.8	0.8	2	3	4
1	3	0.8	0.4	2	2	2
	4	0.5	0.9	2	2	3
	5	0.4	0.9	2	2	2
	1	0.7	0.4	2	2	2
	2	0.9	0.4	2	2	2
2	3	0.9	0.4	2	2	2
	4	0.8	0.6	2	2	3
	5	0.8	0.8	2	3	4

shows two diagnostic generators $p_1,p_2$ which define all considered diagnostic granules $p=\langle s,f\rangle$ belonging to n-ScIFS using the operations described in Formulas (S1)–(S5).

The number n satisfying the Definition of n-ScIFS is searched for as follows.

For the diagnostic generator p, each pair $(\mu ,v)$ has function values ${\mu }_p,v_p$. Then, the smallest natural number n is searched, such that for each $x\in X$:

$$\begin{array}{c}0\le S{c}_n^{-1}\left({\mu }_p\left(x\right)\right)+S{c}_n^{-1}\left(v_p\left(x\right)\right)\le 1,\end{array}$$

(10)

$$\begin{array}{c}0\le S{c}_n\left({\mu }_p\left(x\right)\right)+S{c}_n\left(v_p\left(x\right)\right)\le 1.\end{array}$$

(11)

Then, $(\mu ,v)\in \mathit{n-ScIFS}$.

From among the numbers determined in this way, n is selected as the largest, for all diagnosed objects $o\in X$. The largest number n for the map scale is 2, for the logarithmic scale it is 3, and for the power scale it is 4. According to Theorem 5, for the map scale $p\in \mathit{n-ScIFS}$, when $n>1$, for the logarithmic scale $p\in \mathit{n-ScIFS}$ when $n>2$, and for the power scale $p\in \mathit{n-ScIFS}$, when $n>3$.

6.3. Examples of Calculations on Granules, Their Order, and Elements

On the basis of axioms (6) and (7) of Definition 10 and conditions S2 and S3 in the standard diagnostic granules system, the following calculation formulas are adopted for the diagnostic generators $p_1,p_2$:

S2a.$\left(p_1{\cup }_G{p}_2\right)\left(o\right)=$$(p_1\oplus p_2)\left(o\right)=$$\langle {\mu }_{p1}\left(o\right),v_{p1}\left(o\right)\rangle \oplus \langle {\mu }_{p2}\left(o\right),v_{p2}\left(o\right)\rangle =$$\langle max\{u_{p1}\left(o\right),u_{p2}\left(o\right)\},min\{v_{p1}\left(o\right),v_{p2}\left(o\right)\}\rangle ,$

S3a.$\left(p_1{\cap }_G{p}_2\right)\left(o\right)=(p_1\otimes p_2)\left(o\right)=$$\langle {\mu }_{p1}\left(o\right),v_{p1}\left(o\right)\rangle \otimes \langle {\mu }_{p2}\left(o\right),v_{p2}\left(o\right)\rangle =$$\langle min\{u_{p1}\left(o\right),u_{p2}\left(o\right)\},max\{v_{p1}\left(o\right),v_{p2}\left(o\right)\}\rangle .$

The calculation results are presented in

Table 2. Calculation of $(p_1\oplus p_2)\left(o\right)=\langle s,f\rangle$ and $(p_1\otimes p_2)\left(o\right)=\langle s,f\rangle$ for the diagnostic generators $p_1,p_2=\langle s,f\rangle$.

Object	${\mathit{p}}_1\left(\mathit{o}\right)$		${\mathit{p}}_2\left(\mathit{o}\right)$		$({\mathit{p}}_1\oplus {\mathit{p}}_2)\left(\mathit{o}\right)$		$({\mathit{p}}_1\otimes {\mathit{p}}_2)\left(\mathit{o}\right)$
$\mathit{O}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$
1	0.5	0.4	0.7	0.4	0.7	0.4	0.5	0.4
2	0.8	0.8	0.9	0.4	0.9	0.4	0.8	0.8
3	0.8	0.4	0.9	0.4	0.9	0.4	0.8	0.4
4	0.5	0.9	0. 8	0.6	0.8	0.6	0.5	0.9
5	0.4	0.9	0.8	0.8	0.8	0.8	0.4	0.9

.

According to Table 2, $p_1{\cup }_G{p}_2=p_2$ and $p_1{\cap }_G{p}_2=p_1$. This is also due to the fact that $p_1{\subseteq }_G{p}_2$, i.e., $p_1\left(o\right){\le }_n{p}_2\left(o\right)$ iff $\langle {\mu }_{p1}\left(o\right),v_{p1}\left(o\right)\rangle {\le }_n\langle {\mu }_{p2}\left(o\right),v_{p2}\left(o\right)\rangle$ iff ${\mu }_{p1}\left(o\right)\le {\mu }_{p2}\left(o\right)$ and $v_{p2}\left(o\right)\le v_{p1}\left(o\right)$.

For example, $p_1\left(4\right){\le }_n{p}_2\left(4\right)$ iff $\langle {\mu }_{p1}\left(4\right),v_{p1}\left(4\right)\rangle {\le }_n\langle {\mu }_{p2}\left(4\right),v_{p2}\left(4\right)\rangle$ iff $\langle 0.5,0.9\rangle {\le }_n\langle 0.8,0.6\rangle$ iff $0.5\le 0.8$ and $0.6\le 0.9$.

Indeed, if ${\mu }_{p1}\left(o\right)\le {\mu }_{p2}\left(o\right)$ and $v_{p2}\left(o\right)\le v_{p1}\left(o\right)$, then from Formulas (S2a) and (S3a) we get:

• $\left(p_1{\cup }_G{p}_2\right)\left(o\right)=\langle max\{u_{p1}\left(o\right),u_{p2}\left(o\right)\},min\{v_{p1}\left(o\right),v_{p2}\left(o\right)\}\rangle =\langle {\mu }_{p2}\left(o\right),v_{p2}\left(o\right)\rangle =p_2\left(o\right)$,
• $\left(p_1{\cap }_G{p}_2\right)\left(o\right)=\langle min\{u_{p1}\left(o\right),u_{p2}\left(o\right)\},max\{v_{p1}\left(o\right),v_{p2}\left(o\right)\}\rangle =\langle {\mu }_{p1}\left(o\right),v_{p1}\left(o\right)\rangle =p_1\left(o\right).$

The above formulas can also be analyzed in the context of determining the singletons of granules contained in the generators of granules or by examining the elements of these generators;

Table 3. Example singletons $e_1,e_2,e_3=\langle s,f\rangle$ contained in the diagnostic generators $p_1=\langle s,f\rangle$.

Object	${\mathit{p}}_1\left(\mathit{o}\right)$		${\mathit{e}}_1\left(\mathit{o}\right)$		${\mathit{e}}_2\left(\mathit{o}\right)$		${\mathit{e}}_3\left(\mathit{o}\right)$
$\mathit{O}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$
1	0.5	0.4	0	1	0	1	0.5	0.6
2	0.8	0.8	0	1	0	1	0	1
3	0.8	0.4	0.8	0.4	0.7	0.7	0	1
4	0.5	0.9	0	1	0	1	0	1
5	0.4	0.9	0	1	0	1	0	1

shows selected singletons contained in generator $p_1$.

If $p_1\left(o_i\right)=\langle {\mu }_{p1}\left(o_i\right),v_{p1}\left(o_i\right)\rangle \ne \langle 0,1\rangle$, then $e\left(o_i\right)=\langle {\mu }_e\left(o_i\right),v_e\left(o_i\right)\rangle {\le }_n\langle {\mu }_{p1}\left(o_i\right),v_{p1}\left(o_i\right)\rangle$ and $e\left(o_i\right)=\langle 0,1\rangle$ for $o\ne o_i$.

Thus e is a singleton contained in p: $e\in G_0,e{\subseteq }_G{p}_1$.

Hence $\langle o_i,\langle {\mu }_e\left(o_i\right),v_e\left(o_i\right)\rangle \rangle {\in }_G{p}_1.$

For example: $\langle 3,\langle {\mu }_{e2}\left(3\right),v_{e2}\left(3\right)\rangle \rangle =\langle 3,\langle 0.7,0.7\rangle \rangle {\in }_G{p}_1\left(3\right)=\langle 0.8,0.4\rangle$. Moreover $e_2{\subseteq }_G{e}_1$.

6.4. Calculation of the Repeatability of Diagnostic Granules

Repeatability of diagnostic granules in the map scale

$S{c}_n\left(x\right)=x/n,S{c}_n^{-1}\left(x\right)=nx,$ for $n>0,x\in [0,1],\langle s,f\rangle \in \mathit{n-ScIFS}$ (Example 3).

$\lambda \langle s,f\rangle =\langle S{c}_n^{-1}\left(\lambda S{c}_n\left(s\right)\right),S{c}_n^{-1}(1-\lambda (1-S{c}_n\left(f\right)))\rangle =\langle n\lambda s/n,n(1-\lambda (1-f/n))\rangle =\langle \lambda s,n-n\lambda +\lambda f\rangle$,

${\langle s,f\rangle }^{\lambda }=\langle S{c}_n^{-1}(1-\lambda (1-S{c}_n\left(s\right))),S{c}_n^{-1}\left(\lambda S{c}_n\left(f\right)\right)\rangle =\langle n(1-\lambda (1-fsn)),n\lambda s/n\rangle =\langle n-n\lambda +\lambda s,\lambda f\rangle$.

Repeatability of diagnostic granules in the logarithmic scale.

$S{c}_n\left(x\right)=1-{log}_{(1-1/n)}x,S{c}_1^{-1}\left(x\right)=x$ and for $n>1,S{c}_n^{-1}\left(x\right)=(1-1/n)$^x, for $n>1,x\in (0,1]$, when $S{c}_1\left(x\right)=x$ and $S{c}_n\left(0\right)=0,\langle s,f\rangle \in \mathit{n-ScIFS}$ (Example 4), where a^$x{=}_{df}{a}^x,$

$\lambda \langle s,f\rangle =\langle S{c}_n^{-1}\left(\lambda S{c}_n\left(s\right)\right),S{c}_n^{-1}(1-\lambda (1-S{c}_n\left(f\right)))\rangle =\langle (1-1/n)$ ^ $\lambda (1-{log}_{(1-1/n)}s),(1-1/n)$ ^ $(1-\lambda (1–1+{log}_{(1-1/n)}f))\rangle =\langle ((1-1/n)$ ^ $\lambda )/s$ ^ $\lambda ,(1-1/n)$ ^ $(1-{log}_{(1-1/n)}(f$ ^ $\lambda \left)\right))\rangle =\langle ((1-1/n)/s)$ ^ $\lambda ,(1-1/n)/(f$ ^ $\lambda )\rangle$

${\langle s,f\rangle }^{\lambda }=\langle S{c}_n^{-1}(1-\lambda (1-S{c}_n\left(s\right))),S{c}_n^{-1}\left(\lambda S{c}_n\left(f\right)\right)\rangle =\langle (1-1/n)/(s$ ^ $\lambda ),((1-1/n)/f)$ ^ $\lambda \rangle .$

Repeatability of diagnostic granules in the power scale

$S{c}_n\left(x\right)=x^n,S{c}_n^{-1}\left(x\right)=x^{1/n}$, for $n>0,x\in [0,1],\langle s,f\rangle \in \mathit{n-ScIFS}$ (Example 5).

$\lambda \langle s,f\rangle =\langle S{c}_n^{-1}\left(\lambda S{c}_n\left(s\right)\right),S{c}_n^{-1}(1-\lambda (1-S{c}_n\left(f\right)))\rangle =\langle {\left(\lambda s^n\right)}^{1/n},{(1-\lambda (1-f^n))}^{1/n}\rangle =\langle {\lambda }^{1/n}s,{(1-\lambda (1-f^n))}^{1/n}\rangle ,$

${\langle s,f\rangle }^{\lambda }=\langle S{c}_n^{-1}(1-\lambda (1-S{c}_n\left(s\right))),S{c}_n^{-1}\left(\lambda S{c}_n\left(f\right)\right)\rangle =\langle {(1-\lambda (1-s^n))}^{1/n},{\lambda }^{1/n}f\rangle .$

It is easy to see that for $\lambda =1,\lambda \langle s,f\rangle ={\langle s,f\rangle }^{\lambda }=\langle s,f\rangle$ and for $\lambda =0,\lambda \langle s,f\rangle =\langle 0,1\rangle ,{\langle s,f\rangle }^{\lambda }=\langle 1,0\rangle$.

Furthermore, in

Table 4. Calculations of linear repeatability of granules in the power scale: $\lambda p_1$.

			$\mathit{\lambda }$
Object	${\mathit{p}}_{\mathit{1}}$		1		0.8		0.6		0.4		0.2		0
$\mathit{o}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$
1	0.50	0.40	0.50	0.40	0.47	0.69	0.44	0.80	0.40	0.88	0.33	0.95	0	1
2	0.80	0.80	0.80	0.80	0.76	0.85	0.70	0.90	0.64	0.93	0.53	0.97	0	1
3	0.80	0.40	0.80	0.40	0.76	0.69	0.70	0.80	0.64	0.88	0.53	0.95	0	1
4	0.50	0.90	0.50	0.90	0.47	0.92	0.44	0.94	0.40	0.96	0.33	0.98	0	1
5	0.40	0.90	0.90	0.40	0.38	0.92	0.35	0.94	0.32	0.96	0.27	0.98	0	1

,

Table 5. Calculations of linear repeatability of granules in the power scale: $\lambda p_2$.

			$\mathit{\lambda }$
Object	${\mathit{p}}_{\mathbf{2}}$		1		0.8		0.6		0.4		0.2		0
$\mathit{o}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$
1	0.70	0.40	0.70	0.40	0.66	0.69	0.62	0.80	0.56	0.88	0.47	0.95	0	1
2	0.90	0.40	0.90	0.40	0.85	0.69	0.79	0.80	0.72	0.88	0.60	0.95	0	1
3	0.90	0.40	0.90	0.40	0.85	0.69	0.79	0.80	0.72	0.88	0.60	0.95	0	1
4	0.80	0.60	0.80	0.60	0.76	0.74	0.70	0.83	0.64	0.90	0.53	0.95	0	1
5	0.80	0.80	0.80	0.80	0.76	0.85	0.70	0.90	0.64	0.93	0.53	0.97	0	1

,

Table 6. Calculations of the power repeatability of granules in the power scale: $p_1^{\lambda }$.

			$\mathit{\lambda }$
Object	${\mathit{p}}_{\mathit{1}}$		1		0.8		0.6		0.4		0.2		0
$\mathit{o}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$
1	0.50	0.40	0.50	0.40	0.71	0.38	0.81	0.35	0.89	0.32	0.95	0.27	1	0
2	0.80	0.80	0.80	0.80	0.85	0.76	0.90	0.70	0.93	0.64	0.97	0.53	1	0
3	0.80	0.40	0.80	0.40	0.85	0.38	0.90	0.35	0.93	0.32	0.95	0.27	1	0
4	0.50	0.90	0.50	0.90	0.71	0.85	0.81	0.79	0.89	0.72	0.98	0.60	1	0
5	0.40	0.90	0.40	0.90	0.69	0.85	0.80	0.79	0.88	0.72	0.98	0.60	1	0

and

Table 7. Calculations of the power repeatability of granules in the power scale: $p_2^{\lambda }$.

			$\mathit{\lambda }$
Object	${\mathit{p}}_{\mathbf{2}}$		1		0.8		0.6		0.4		0.2		0
$\mathit{o}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$	$\mathit{s}$	$\mathit{f}$
1	0.70	0.40	0.70	0.40	0.79	0.38	0.86	0.35	0.91	0.32	0.95	0.27	1	0
2	0.90	0.40	0.90	0.40	0.92	0.38	0.94	0.35	0.96	0.32	0.95	0.27	1	0
3	0.90	0.40	0.90	0.40	0.92	0.38	0.94	0.35	0.96	0.32	0.95	0.27	1	0
4	0.80	0.60	0.80	0.60	0.85	0.57	0.90	0.53	0.93	0.48	0.95	0.40	1	0
5	0.80	0.80	0.80	0.80	0.85	0.76	0.90	0.71	0.93	0.64	0.97	0.53	1	0

, it is assumed that the diagnosis can be repeated a maximum of 5 times until certainty is achieved, i.e., $\lambda =1-i/5$, where i is the number of repetitions, i.e., $\lambda =1,4/5,3/5,2/5,1/5,0.$

Table 4 and Table 5 present the results of the calculation of the linear repeatability $\lambda p_1,\lambda p_2$ in the power scale $S{c}_n\left(x\right)=x^n$:

• A decrease in the repeatability degree $\lambda$ leads to a decrease of the symptom degree s,
• A decrease in the repeatability degree $\lambda$ leads to an increase of the faults degree f,
• $\lambda p_1{\subseteq }_G\lambda p_2$ according to the condition (5) of Theorem 10, because $p_1{\subseteq }_G{p}_2$.

Table 6 and Table 7 present the results of the calculations of the power repeatability of $p_1^{\lambda },p_2^{\lambda }$ on the power scale $S{c}_n\left(x\right)=x^n$:

• A decrease in the repeatability degree $\lambda$ leads to an increase of the symptom s degree,
• A decrease in the repeatability degree $\lambda$ leads to a decrease of the faults f degree,
• $p_1^{\lambda }{\subseteq }_G{p}_2^{\lambda }$ according to the condition (5) of Theorem 10, because $p_1{\subseteq }_G{p}_2$.

7. Conclusions

The paper formulates the following procedures:

• Setting the sequences $\left\{S{c}_n\right\},\left\{S{c}_n^{\prime }\right\}$ of functions scaling the values of symptom attributes and fauts, such that from a certain number n there is $0\le S{c}_n\left(\mu \right)+S{c}_n^{\prime }\left(v\right)\le 1$, for any results of diagnostic measurements $\mu ,v\in [0,1]$,
• Determining the number n of the sequences $\left\{S{c}_n\right\},\left\{S{c}_n^{\prime }\right\}$ which satisfy the condition specified in the above procedure 1.

The language of diagnosis was defined as the language of the diagnosis granule system.

Moreover, it was shown that the diagnosis granule system (Definition 13) is induced by the diagnosis granule base (Definition 12). The model of such a system is the standard diagnostic granule system described in this paper. Models of other granule systems have been described in [20,42,43,45,46]. These papers also use many ideas contained in the information granulation literature on the granulation of information.

After years of theoretical work on information granulation, it is correct to assume that information granules are defined by some general axioms of the information granule database (Definition 12). Research on the granulation of self-diagnosis of information retrieval on the Web is also being developed and detailed, as well as in the field of heuristic methods of identifying information, such as analysis, synthesis, reduction, and induction [47]. Furthermore, the application of the theory of the diagnostic granules presented in this paper, implemented as part of the project for the diagnosis of the reliability of the technical objects used in everyday life, will allow for the proper selection of scaling functions and triangular norms as well as standards of repetition of the diagnosis of these objects. This paper is both an elaboration of the syntax and semantics of the descriptive language of diagnosis, i.e., the system of diagnosis granules. For example, this language will make it possible to represent knowledge about cybersecurity of information retrieval on the web.