Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques
11 Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques
S. R. Kannan and Rajesh Kumar Mohapatra
1, * Pondicherry University (A Central University of India), Puducherry, India * Corresponding author: [email protected]
Abstract.
The novelty of this paper is to construct the explicit combinatorial formula for the number of all distinct fuzzy matrices of finite order, which leads us to invent a new sequence. In order to achieve this new sequence, we analyze the behavioral study of equivalence classes on the set of all fuzzy matrices of a given order under a suitable natural equivalence relation. In addition this paper characterizes the properties of non-equivalent classes of fuzzy matrices of order ๐ with elements having degrees of membership values anywhere in the closed unit interval [0,1] . Further, this paper also derives some important relevant results by enumerating the number of all distinct fuzzy matrices of a given order in general. And also, we achieve these results by incorporating the notion of ๐ -level fuzzy matrices, chains, and flags (maximal chains). Keywords:
Fuzzy matrices; ๐ -level fuzzy matrices; Chains; Flags; Binomial numbers One of the most challenging problems of fuzzy matrix theory is to classify for concerning the study of the set of all fuzzy matrices of any given order and to count the same. Counting of all distinct fuzzy matrices of finite order is great of an interest in both and theoretical and practical point of view in contexts of mathematics. In recent years, this topic has undergone a remarkable development in many areas and many interesting important results have been proposed. The pioneering work of Zadeh [1] on fuzzy subsets of a set and Rosenfeld [2] on fuzzy subgroups of a group led to the fuzzification of some algebraic structures like group, ring, etc. Though handling such a problem is quite a difficult job, many researchers have been still working on classifying and counting both fuzzy subsets and fuzzy subgroups in the last few years. Research works on counting the number of fuzzy subsets of a finite set was initially noticed by Murali and Makamba [3] and later many other researchers such as Tฤrnฤucean [4], ล eลกelja and Tepavฤeviฤ [5], and Jain [6] have taken further away. To determine the number of fuzzy subgroups of finite groups, at first, several papers have treated the special cases of finite abelian groups such as computing the number of distinct square-free order of fuzzy cyclic group [3], fuzzy cyclic group of order ๐ ๐ ๐ ๐ ( ๐, ๐ primes) ([7], [8]โ[11]) . Later, the authors in ([4], [12]) deal for determining the number of distinct fuzzy subgroups for two classes of finite abelian groups: finite cyclic groups and finite elementary abelian ๐ -groups. Further, this investigation has been propagated to some remarkable classes of non-abelian groups: symmetric groups ([13], [14]), alternating group [15], dihedral groups ([16], [17], [18]), hamiltonian groups [19]. Subsequently, the same problems were also implemented and analyzed in the case of the fuzzy normal subgroup ([20], [16]). The research works on calculating the formula for the number of all distinct fuzzy matrices through a combinatorial approach are very important in deriving a new sequence that can be used in many real-life applications. Researchers have concentrated their research activities on counting the fuzzy subsets of a finite set as well as fuzzy subgroups of a finite group, but no researchers have focused their research works on computing the number of fuzzy matrices. Since the problem of counting the number of non-equivalent fuzzy subsets in one-dimensional space was obtained, but it remains still open for two-dimensional space, i.e., the formula for calculating the number of non-equivalent classes of fuzzy matrices. Thus, it motivates us to contribute novel research for invention in the field of fuzzy matrix theory ([21], [22]). In order to achieve the new invents on establishing the formula for the number of all distinct fuzzy matrices on finite set, this paper discusses equivalence classes on fuzzy matrices on a finite set under a natural equivalence relation. Because of the fact that fuzzy matrices on a finite set are more abundant than crisp matrices. The necessity of classification of fuzzy matrices have done based on ๐ผ -cuts due to the following fact: there are uncountably many fuzzy matrices on the same domain, even if the domain is either a singleton (or finite) or countable. Since there are different notions of equivalence ([4], [5], [11], [23]โ[27]), research works have grown to be a major branch which have many more interesting challenges in the classification of fuzzy subsets, fuzzy subgroups, etc. The present paper utilizes equivalence [24] by accepting the property on support which is significant to construct the formula for the number of all distinct fuzzy matrices. In this paper we mainly focus to propose a closed explicit formula by indicating the number of all distinct fuzzy matrices of order ๐ . This closed formula leads us to invent a new important sequence that is not present in the Online Encyclopedia of Integer Sequences (OEIS) [28]. The structure of this paper is organized as follows: Section 2 focuses on a few important definitions, results and notational set up which are necessary throughout the paper. Precisely, we will concentrate on counting the number of all fuzzy matrices up to Muraliโs equivalence relation and will derive some of its relevant results in Section 3. In the final section conclusions and further research of this paper are specified.
2. Preliminaries
The main goal of this section is to set up notations, collect some basic definitions and results with its properties to introduce the new important results on counting some specific fuzzy matrices.
Throughout this paper, let us assume that
๐ดฬ = (๐ ๐๐ ) be an ๐ ร ๐ fuzzy matrix with elements having membership values in the real unit interval ๐ผ = [0,1] , where ๐ is a non-zero, non-negative integer. The union ( โช ), intersection ( โฉ ) of two fuzzy matrices, and complementation ( c ) of a fuzzy matrix are defined by using supremum (sup or max) and infimum (inf or min) component-wise, and ๐๐ operator pointwise, respectively [1], [29]. The containment of a fuzzy matrix ๐ดฬ in a fuzzy matrix ๐ตฬ , denoted as ๐ดฬ โ ๐ตฬ if ๐ ๐๐ โค ๐ ๐๐ for all ๐, ๐ . Further we denote fuzzy matrices by ๐ดฬ, ๐ตฬ , ๐ถฬ, etc. and its membership values by ๐ผ, ๐ฝ, ๐พ, etc. Through an ๐ผ -cut of fuzzy matrix ๐ดฬ for ๐ผ belongs to ๐ผ , we have a crisp matrix ๐ดฬ ๐ผ = { 1, ๐ ๐๐ โฅ ฮฑ0, otherwise . This is called the weak ๐ผ -cut . By strong ๐ผ -cut we mean ๐ดฬ ฮฑ = {1, if ๐ ๐๐ > ฮฑ and otherwise } . But, in this paper we are always dealt with weak ๐ผ -cut. It is easy to verify that for โฒ โค 1 , we have ๐ดฬ ๐ผ โ ๐ดฬ ฮฑ โฒ . For any arbitrary fuzzy matrix ๐ดฬ it can be decomposed into a union of characteristic function as follows: Theorem 2.1 [30], [29], [31] . For any fuzzy matrix
๐ดฬ = โช ๐ผ {๐ผ๐ ๐ดฬ ๐ผ : 0 โค ๐ผ โค 1} , where ๐ ๐ดฬ ๐ผ denotes the characteristic function of the crisp matrix ๐ดฬ ๐ผ . To prove our results, we recall some known definitions and theorems in the area of combinatorics.
Definition 2.2 ( Binomial Number ) [32] . Let us denote ( ๐๐ ) be the number of ๐ -element subsets of an ๐ -element set; that is nothing but the number of ways we can select ๐ distinct elements from an ๐ -element set. This is well-known as a binomial number or a binomial coefficient . ( An alternative notation,
๐ถ(๐, ๐) ) Now we are going to present a famous theorem, known as binomial theorem.
Theorem 2.3 ( Binomial Theorem ) [32] . For all integers ๐ โฅ 0 , (๐ + ๐) ๐ = โ ( ๐๐ ) ๐ ๐ ๐๐=0 ๐ ๐โ๐ . In fact, one has to give the familiar formula of binomial numbers. (By definition,
0! = 1 .) Lemma 2.4.
For ๐ โฅ ๐ โฅ 0 , ( ๐๐ ) = ๐!(๐โ๐)! , with the convention that ( ๐๐ ) = 0 , for any ๐ > ๐ . Equivalent fuzzy matrices and concepts of chains
The purpose of this section is to briefly discuss the study of an equivalence relation on the set of all fuzzy matrices and the concept of chains. Thus we start with an equivalence relation โ defined on any class of fuzzy matrices as follows: Definition 2.5 [24], [33], [34] . ๐ดฬ โ ๐ตฬ if and only if for all ๐, ๐, ๐, ๐ (i) ๐ ๐๐ > ๐ ๐๐ if and only if ๐ ๐๐ > ๐ ๐๐ (ii) ๐ ๐๐ = 1 if and only if ๐ ๐๐ = 1 (iii) ๐ ๐๐ = 0 if and only if ๐ ๐๐ = 0 . It is easy to verify that this relation is indeed an equivalence relation on the set of all fuzzy matrices and when its entries restricted to ๐ผ โฒ = {0, 1}, which corresponds with equality of crisp matrices. Based on this equivalence relation, equivalence class having ๐ดฬ is denoted as [๐ดฬ] and two fuzzy matrices ๐ดฬ and ๐ตฬ are distinct if ๐ดฬ and ๐ตฬ are not equivalent, that is, ๐ดฬ โ ๐ตฬ . The next proposition proposes the relation between ๐ผ -cuts and equivalence. Proposition 2.6 [35] . Let ๐ดฬ and ๐ตฬ be two fuzzy matrices of order ๐ . Then ๐ดฬ โ ๐ตฬ if and only if for each ๐ผ > 0 there exists a ๐ฝ > 0 such that ๐ดฬ ๐ผ = ๐ตฬ ๐ฝ . It follows from the above proposition that the equivalent fuzzy matrices can be analyzed by their ๐ผ -cuts. This observation promotes us an innovation to raise the ideas ๐ -level fuzzy matrices, chains, and flags. Now we will elaborate them explicitly in the following subsection. ๐ -level fuzzy matrices and flags We will classify non-equivalent classes of fuzzy matrices by incorporating the concept ๐ผ -cuts for . In this subsection, we briefly give some definitions using the basic ideas. Definition 2.7.
For ๐ โ โ and , by a ๐ -level fuzzy matrix ๐ดฬ , it means ๐ดฬ has ๐ -number of distinct membership values in the open unit interval, that is, ๐ผ \ {0,1} . Explicitly, a ๐ -level fuzzy matrix of order ๐ is a (๐ + 1) -pair of a chain of crisp matrices under the usual inclusion of the form ฮ โถ ๐ดฬ ๐ผ โ ๐ดฬ ๐ผ โ ๐ดฬ ๐ผ โ โฏ โ ๐ดฬ ๐ผ ๐ with > ๐ผ > ๐ผ > โฏ > ๐ผ ๐ โฅ 0 ฮฑ ๐ โs are in the unit interval ๐ผ , and not necessarily including and , written in the descending order of magnitude. Then a fuzzy matrix ๐ดฬ = โช ๐ผ ๐ {๐ผ ๐ ๐ ๐ดฬ ๐ผ๐ : 0 โค ๐ผ ๐ โค 1} has its ๐ผ ๐ -cuts equals ๐ดฬ ๐ผ ๐ . Here ๐ดฬ ๐ผ ๐ โs are called various components of the chain ฮ . It follows that the above inclusions in the chain ฮ are always taken to be strict. By taking ๐ = ๐ in the above chain ฮ , we can get a maximal chain, that is named as a flag. We call two ๐ -level fuzzy matrices are distinct if they are not equivalent. It is also clear that for any fuzzy matrix of order ๐ can have maximum ๐ -distinct values of membership degree in ๐ผ . Hence the maximum possible number of distinct ๐ผ -cut relational matrices of a fuzzy matrix of order ๐ is ๐ + 1 . The cause behind is that the set of ๐ -distinct real numbers in the open interval (0,1) will split the closed interval [0,1] into ๐ + 1 segments. Now take an ๐ผ in each open segment will provide a distinct ๐ผ -cut. Therefore, the length of the flag for a fuzzy matrix of order ๐ is ๐ + 1 . Hence, it can be concluded that ๐ดฬ โ ๐ตฬ if and only if ๐ดฬ and ๐ตฬ determine the same chain of crisp matrices of type ฮ . It obviously follows that there is a one-to-one correspondence between the set of ๐ -level distinct fuzzy matrices of order ๐ and the set of chains crisp matrices of order ๐ of length ๐ under the usual inclusion. Definition 2.8.
Let ๐ be a positive integer and let ๐ด , ๐ด , ๐ด , โฆ ๐ด ๐ be crisp matrices of order ๐ with ๐ด ๐ โ ๐ด ๐+1 for ๐ = 0, ๐ โ 1ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ with then the following matrix chain of the type ๐ค โถ ๐ด โ ๐ด โ ๐ด โ โฏ โ ๐ด ๐ is called a chain of crisp matrices of order ๐ . In this case, the integer ๐ is called the length of the proper subgroup chain ๐ค , and the subgroups ๐ด and ๐ด ๐ are called the initial term and the terminal term of ๐ค . Definition 2.9. (i) Let ๐ be an ๐ ร ๐ null matrix is a matrix that is defined as ๐ดฬ = (๐ ๐๐ ) ๐ร๐ , where ๐ ๐๐ = 0 for all ๐, ๐ โ { 1, 2, โฆ , ๐} and ๐ฝ be a unit matrix of order ๐ is a matrix which is defined as ๐ฝ = (๐ ๐๐ ) ๐ร๐ , where ๐ ๐๐ = 1 for every . (ii) A chain of crisp matrices of order ๐ is called ๐ -rooted (respectively, ๐ฝ -rooted) if it contains ๐ (respectively, ๐ฝ ). Otherwise, it is simply called a chain of crisp matrices of order ๐ . It can be remark that there is a one-to-one correspondence between the set of ๐ -rooted (respectively, ๐ฝ -rooted) ๐ -level distinct fuzzy matrices of order ๐ and the set of ๐ -rooted (respectively, ๐ฝ -rooted) chains of crisp of matrices of order ๐ of length ๐ under inclusion. Definition 2.10.
Suppose ๐ be a positive integer and . Let โณ be the set of all crisp matrices of order ๐ and let ๐ถ be the family of all chains of crisp matrices of order ๐ of length ๐ . Now set ๐น๐ ๐,๐ (โณ) = {๐ค โ ๐ถ | the length of ๐ค is ๐ , that is the initial term and the terminal term of ๐ค are not necessarily be the null matrix and the unit matrix respectively}. ๐น๐ ๐,๐๐ (โณ) = {๐ค โ ๐ถ | the initial term of ๐ค is ๐ด = ๐ of length ๐} ; ๐น๐ ๐,๐๐ฝ (โณ) = {๐ค โ ๐ถ | the terminal term of ๐ค is ๐ด ๐ = ๐ฝ of length ๐} ; Let us denote ๐น๐ ๐,๐ (โณ), ๐น๐ ๐,๐๐ (โณ) and ๐น๐ ๐,๐๐ฝ (โณ) be the set of all chains of crisp matrices of order ๐ of length ๐ , ๐ -rooted chains of crisp matrices of order ๐ of length ๐ and ๐ฝ -rooted chains of crisp matrices of order ๐ of length ๐ , respectively. We use notation ๐ ๐,๐ , ๐ ๐,๐๐ and ๐ ๐,๐๐ฝ to denote the cardinal numbers of ๐น๐ ๐,๐ (โณ), ๐น๐ ๐,๐๐ (โณ) and ๐น๐ ๐,๐๐ฝ (โณ) respectively. The relations between these numbers are of the following. Remark 2.11.
It is obvious that ๐น๐ ๐,๐๐ (โณ) โ ๐น๐ ๐,๐ (โณ) and ๐น๐ ๐,๐๐ฝ (โณ) โ ๐น๐ ๐,๐ (โณ) . Hence, ๐ ๐,๐๐ < ๐ ๐,๐ and ๐ ๐,๐๐ฝ < ๐ ๐,๐ . Also, we denote ๐น๐ ๐ (โณ), ๐น๐ ๐๐ (โณ) and ๐น๐ ๐๐ฝ (โณ) be the set of all chains of crisp matrices of order ๐ , ๐ -rooted chains of crisp matrices of order ๐ and ๐ฝ -rooted chains of crisp matrices of order ๐ , respectively. Take ๐ ๐ = |๐น๐ ๐ (โณ)|, ๐ ๐๐ = |๐น๐ ๐๐ (โณ)| and ๐ ๐๐ฝ = |๐น๐ ๐๐ฝ (โณ)| . Similarly, we can conclude that there is also a one-to-one correspondence between the collection of distinct fuzzy matrices of order ๐ (respectively, ๐ -rooted distinct fuzzy matrices of order ๐ , ๐ฝ -rooted distinct fuzzy matrices of order ๐ ) and the collection of chains crisp matrices of order ๐ (respectively, ๐ -rooted chains crisp matrices of order ๐ , ๐ฝ -rooted chains crisp matrices of order ๐ ) under inclusion.
3. Enumeration of fuzzy matrices
In this section, we keen to develop the method for counting the number of
๐น๐(โณ) . Here, the explicit closed combinatorial summation formulae were discovered for computing the number of ๐น๐ ๐ (โณ), ๐น๐ ๐๐ (โณ) and ๐น๐ ๐๐ฝ (โณ) . In order to determine these numbers, we first establish the number of ๐น๐ ๐,๐ (โณ), ๐น๐ ๐,๐๐ (โณ) and ๐น๐ ๐,๐๐ฝ (โณ) in the following section. ๐ -level fuzzy matrices This subsection motivates to achieve our objective of finding the formulas for calculating the numbers ๐ ๐,๐ , ๐ ๐,๐๐ and ๐ ๐,๐๐ฝ . In order to derive the explicit formula of ๐ ๐,๐ , we shall first investigate with some of its specific cases. Let us take ๐ = 2 , the lattice of ๐น๐ (โณ) ( that is the set of all crisp matrices of order (โณ)) is constituted by the following matrices: ๐ด = ( ) , ๐ด = ( ) , ๐ด = ( ) , ๐ด = ( ) , ๐ด = ( ) , ๐ด = ( ) , ๐ด = ( ) , ๐ด = ( ) , ๐ด = ( ) , ๐ด = ( ) , ๐ด = ( ) , ๐ด = ( ) , ๐ด = ( ) , ๐ด = ( ) , ๐ด = ( ) , ๐ด = ( ) , and its lattice structure has shown in Figure 1. For ๐ = 2 and ๐ = 3 . We can precisely determine the number of ๐น๐ (โณ) , by describing all chains through manually by direct calculation in the five possibilities as listed below: Figure 1.
Graphical illustration of
๐ฟ(๐น๐ (โณ))
Case I.
Consider the chains in ๐น๐ (โณ) of the type ๐ด โ ๐ด โ ๐ด โ ๐ด have the following chains: ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด . Case II.
The chains in ๐น๐ (โณ) of the type ๐ด โ ๐ด โ ๐ด โ ๐ด , have the following chains: ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด . Case III.
The chains in ๐น๐ (โณ) of the type ๐ด โ ๐ด โ ๐ด โ ๐ด , have the following chains: ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด . Case IV.
The chains in ๐น๐ (โณ) of the type ๐ด โ ๐ด โ ๐ด โ ๐ด , have the following chains: ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด . Case V.
The chains in ๐น๐ (โณ) of the type ๐ด โ ๐ด โ ๐ด โ ๐ด , have the following chains: ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , Figure 2.
Graphical illustration of ๐น๐ (โณ) . ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด , ๐ด โ ๐ด โ ๐ด โ ๐ด . Thus, it is calculated from the above five cases that the number ๐ is ๐ = 24 + 12 + 12 + 12 + 24 = 84 . For all other particular cases, we will not provide a list of chains to avoid bulkiness. However, the graphical structure of ๐น๐ (โณ) have displayed in Figure 2. The observation through above examples and by analyzing Figure 1 and Figure 2, we come to conclude that the method of direct calculation doesnโt work through manually (or becomes too complex) for other cases. Thus these problems motivate that we must find another method in order to determine the formula of ๐ ๐,๐ . Thus, we give the theorem as follows: Lemma 3.1.
For any ๐ โฅ 1 and ๐ โ {0, 1,2, โฆ , ๐ } , the number of ๐น๐ ๐,๐ (โณ) โs are given by the equality: ๐ ๐,0 =โ ( ๐ ๐ ) ๐ ๐ =0 ; ๐ ๐,1 = โ โ ( ๐ ๐ ) ( ๐ โ๐ ๐ ) ๐ โ๐ ๐ =1๐ โ1๐ =0 ; ๐ ๐,2 = โ โ โ ( ๐ ๐ ) ( ๐ โ๐ ๐ ) ๐ โ๐ โ๐ ๐ =1๐ โ๐ โ1๐ =1๐ โ2๐ =0 ( ๐ โ๐ โ๐ ๐ ) ; โฎ ๐ ๐,๐ = โ โ โฆ โ ( ๐ ๐ ) ( ๐ โ๐ ๐ ) โฆ ๐ โ๐ โ๐ โ๐ โโฏ๐ ๐โ1 โ๐+๐๐ ๐ =1๐ โ๐ โ๐+1๐ =1๐ โ๐๐ =0 ( ๐ โ๐ โ๐ โ๐ โโฏ๐ ๐โ1 ๐ ๐ ) . In particular, for ๐ = ๐ we have the number ๐ ๐,๐ of all distinct flags of crisp matrices of order ๐ is ๐ ๐,๐ = โ (๐ โ๐+1) ๐2โ๐+1 ๐!๐ ๐=1 . Proof.
We start the proof by making an auxiliary construction. It is known that all the crisp matrices of order ๐ are ๐ด , ๐ด , ๐ด , โฆ , ๐ด ๐ , where the number of entries in ๐ด ๐ , |๐ด ๐ | = ( ๐ ๐ ) for ๐ โ โ and ๐ = 0,1,2, โฆ , ๐ . And these crisp matrices satisfy the following condition. ๐ด โ ๐ด โ ๐ด โ โฏ โ ๐ด ๐ , for ๐ = 0, ๐ ฬ ฬ ฬ ฬ ฬ ฬ Now ๐น๐ ๐,0 (โณ) = {๐ด ๐ | 0 โค ๐ โค ๐ } ; ๐น๐ ๐,1 (โณ) = {๐ด ๐ โ ๐ด ๐ |0 โค ๐ < ๐ โค ๐ } ; โฎ ๐น๐ ๐,๐ (โณ) = {๐ด ๐ โ ๐ด ๐ โ ๐ด ๐ โ โฏ โ ๐ด ๐ k | 0 โค ๐ < ๐ < ๐ < โฏ < ๐ ๐ โค ๐ }. Next, our aim to find the cardinality of ๐น๐ ๐,๐ (โณ) for ๐ = 0,1,2, โฆ , ๐ and โ ๐ โ โ . It can easily verify that ๐ ๐,0 = |๐น๐ ๐,0 (โณ)| = 2 ๐ . Let us consider ๐ค be a chain of crisp matrices in ๐น๐ ๐,๐ (โณ) for as follows: ๐ค โถ ๐ด โ ๐ด โ ๐ด โ โฏ โ ๐ด ๐ . Then one can obviously get the following (๐ + 1) -dimensional vector as ๐ผ = (|๐ด |, |๐ด |, |๐ด |, โฆ , |๐ด ๐ |) . For our convenience, let us call ๐ผ the order vector of ๐ค . Now consider ๐บ = { ๐ผ | ๐ผ is an order vector of
๐ค, ๐ค โ ๐น๐ ๐,๐ (โณ)} and for any ๐ผ โ ๐บ, ๐น๐ ๐,๐๐ผ (โณ) = {๐ค โ ๐น๐ ๐,๐ (โณ) | the order of ฮ is ๐ผ} . Then it is very natural to see that ๐น๐ ๐,๐ (โณ) = โ ๐น๐ ๐,๐๐ผ (โณ) ๐ผโ๐บ , and ๐น๐ ๐,๐๐ผ (โณ) โฉ ๐น๐ ๐,๐๐ฝ (โณ) = โ if ๐ผ โ ๐ฝ . Therefore, ๐ ๐,๐ = โ |๐น๐ ๐,๐๐ผ (โณ)| ๐ผโ๐บ It is easy to notice that
๐บ = {๐ผ = (( ๐ ๐ ) , ( ๐ ๐ ) , ( ๐ ๐ ) , โฆ , ( ๐ ๐ ๐ )) | 0 โค ๐ < ๐ < ๐ < โฏ < ๐ ๐ โค ๐ } . For the order vector ๐ผ of ๐ค โ ๐น๐ ๐,๐ (โณ) of length ๐ , the number of choices for ๐ ๐ก term , 2 ๐๐ term , โฏ , ๐ ๐กโ term of vector ๐ผ are ( ๐ ๐ ) , 0 โค ๐ โค ๐ โ ๐ ; ( ๐ โ๐ ๐ ) , 1 โค ๐ โค ๐ โ ๐ โ ๐ + 1 ; โฎ ( ๐ โ๐ โ๐ โ๐ โโฏ๐ ๐โ1 ๐ ๐ ) , 1 โค ๐ ๐ โค ๐ โ ๐ โ ๐ โ ๐ โ โฏ ๐ ๐โ1 . Therefore, we can write exactly the formula ๐ ๐,๐ as follows: ๐ ๐,๐ = โ โ โฆ โ ( ๐ ๐ ) ( ๐ โ๐ ๐ ) โฆ ๐ โ๐ โ๐ โ๐ โโฏ๐ ๐โ1 โ๐+๐๐ ๐ =1๐ โ๐ โ๐+1๐ =1๐ โ๐๐ =0 ( ๐ โ๐ โ๐ โ๐ โโฏ๐ ๐โ1 ๐ ๐ ) . In particular, taking ๐ = ๐ in ๐ ๐,๐ we have ๐ ๐,๐ = โ ( ๐ ๐ ) n ๐=0 = ( ๐ ) ( ๐ ) ( ๐ ) โฆ ( ๐ ๐ ) = (๐ ) ๐2 (๐ โ1) ๐2โ1 (๐ โ2) ๐2โ2 โฆ11!2!3!โฆ๐ ! = โ (๐ โ๐+1) ๐2โ๐+1 ๐!๐ ๐=1 . This completes the proof of the theorem. โ In order to understand and clarify the above Lemma 3.1, we will estimate the number ๐ in the following: Example 3.2.
Find the number ๐ . Solution.
By using the explicit formula ๐ ๐,๐ in Lemma 3.1. We shall find the number ๐ by taking ๐ = 2 and ๐ = 2 . Then ๐ = โ โ โ ( ) ( ๐ ) ( โ๐ ๐ ) โ๐ ๐ =13โ๐ ๐ =12๐ =0 = ( )( )( ) + ( )( )( ) + ( )( )( ) +( )( )( ) + ( )( )( ) + ( )( )( ) + ( )( )( ) + ( )( )( ) + ( )( )( ) + ( )( )( ) =110 . Hence the theorem is clarified. In the following, the first two corollaries are obvious in the view of Lemma 3.1. Corollary 3.3.
For each ๐ โ โ and ๐ = 0,1, โฆ , ๐ , the number of ๐น๐ ๐,๐๐ (โณ) โ s are given by the equality: ๐ ๐,0๐ = 1 ; ๐ ๐,1๐ = โ ( ๐ ๐ ) ๐ ๐ =1 ; ๐ ๐,2๐ = โ โ ( ๐ ๐ ) ( ๐ โ๐ ๐ ) ๐โ๐ ๐ =1๐ โ1๐ =1 ; โฎ ๐ ๐,๐๐ = โ โ โฆ โ ( ๐ ๐ ) ( ๐ โ๐ ๐ ) โฆ ( ๐ โ๐ โ๐ โโฏ๐ ๐โ1 ๐ ๐ ) ๐ โ๐ โ๐ โโฏ๐ ๐โ1 ๐ ๐ =1๐ โ๐ โ๐+2๐ =1๐ โ๐+1๐ =1 . Corollary 3.4.
For any non-zero positive integers ๐ and , the number of ๐น๐ ๐๐ฝ (โณ) โs, are given by the equality: ๐ ๐,0๐ฝ = 1 ; ๐ ๐,1๐ฝ = โ ( ๐ ๐ ) ๐ โ1๐ =0 ; ๐ ๐,2๐ฝ = โ โ ( ๐ ๐ ) ( ๐ โ๐ ๐ ) ๐ โ๐ โ1๐ =1๐ โ2๐ =0 ; โฎ ๐ ๐,๐๐ฝ = โ โ โฆ โ ( ๐ ๐ ) ( ๐ โ๐ ๐ ) โฆ ( ๐ โ๐ โ๐ โโฏ๐ ๐โ1 ๐ ๐ ) ๐ โ๐ โ๐ โโฏ๐ ๐โ1 โ1๐ ๐ =1๐ โ๐ โ๐+1๐ =1๐ โ๐๐ =0 . ๐ ๐ The main objective of this subsection is to expose the closed summation formula ๐ ๐ for the number of all distinct fuzzy matrices of order ๐ . By summing up all ๐ ๐,๐ โ s we can easily obtain an explicit formula for ๐ ๐ . So, the next result follows immediately from Lemma 3.1. Theorem 3.5.
The number ๐ ๐ of all distinct fuzzy matrices of order ๐, ๐น๐ ๐ (โณ) is given by the following equality: ๐ ๐ = โ ๐ ๐,๐๐ ๐=0 =โ (โ โ โฆ โ ( ๐ ๐ ) ( ๐ โ๐ ๐ ) โฆ ๐ โ๐ โ๐ โ๐ โโฏ๐ ๐โ1 โ๐+๐๐ ๐ =1๐ โ๐ โ๐+1๐ =1๐ โ๐๐ =0 ( ๐ โ๐ โ๐ โ๐ โโฏ๐ ๐โ1 ๐ ๐ ) ๐ ๐=0 ) , where ๐ โฅ 1 is arbitrary and fixed. Next, we are going to estimate the total number of terms in the expansion of the closed formula ๐ ๐ in the following proposition. Proposition 3.6.
For any positive integers ๐ โฅ 1 , the number of terms in the expansion of ๐ ๐ is ๐ +1 โ 1 . Proof.
It is well-known that the number of all possible terms in the expression of ๐ ๐,๐ is ( ๐ +1 ๐+1 ) , ๐ = 0,1,2,3, โฆ , ๐ , Therefore, the total number of terms in the expansion of the expression of the formula ๐ ๐ =โ ๐ ๐,๐๐ ๐=0 is ( ๐ +11 ) + ( ๐ +12 ) + โฏ + ( ๐ +1๐ +1 ) = โ ( ๐ +1๐ ) ๐ +1๐=1 , by using Theorem 2.3, we can have โ ( ๐ +1๐ ) ๐ +1๐=1 = 2 ๐ +1 โ 1 . This completes the proof. โ We have the following two immediate straightforward consequence of Theorem 3.5.
Corollary 3.7.
For a fixed value ๐ โ โค + and ๐ โฅ 1 , the number ๐ ๐๐ of all distinct ๐ -rooted fuzzy matrices of order ๐ is given by the following equality: ๐ ๐๐ = โ ๐ ๐,๐๐๐ ๐=0 =โ (โ โ โฆ โ ( ๐ ๐ ) ( ๐ โ๐ ๐ ) โฆ ( ๐ โ๐ โ๐ โโฏ๐ ๐โ1 ๐ ๐ )) ๐ โ๐ โ๐ โโฏ๐ ๐โ1 ๐ ๐ =1๐ โ๐ โ๐+2๐ =1๐ โ๐+1๐ =1๐ ๐=0 . Corollary 3.8.
The number ๐ ๐๐ฝ of all distinct ๐ฝ -rooted fuzzy matrices of order ๐ , where ๐ โฅ 1 , is given by the following equality: ๐ ๐๐ฝ = โ ๐ ๐,๐๐ฝ๐ ๐=0 =โ (โ โ โฆ โ ( ๐ ๐ ) ( ๐ โ๐ ๐ ) โฆ ( ๐ โ๐ โ๐ โโฏ๐ ๐โ1 ๐ ๐ )) ๐ โ๐ โ๐ โโฏ๐ ๐โ1 โ1๐ ๐ =1๐ โ๐ โ๐+1๐ =1๐ โ๐๐ =0๐ ๐=0 . Remark 3.9.
A fuzzy matrix of order , does not have a conventional meaning as there no elements that exist in the matrix. Thus, we simply take ๐ = 1 as an initial condition for the counting function ๐ ๐ for ๐ = 0 . In the following, we construct the table for the number of ๐น๐ ๐,๐ (โณ) , ๐ ๐,๐ (in Lemma 3.1) and the number of ๐น๐ ๐ (โณ) , ๐ ๐ (in Theorem 3.5) for , ๐ โค 3 . Table 1. ๐ ๐ and ๐ ๐,๐ for , ๐ โค 3 nnnnnnnnnnn kn fffffffffffn kf ๏ฝ Thus, we have obtained the number of fuzzy matrices ๐ ๐ for ๐ โฅ 0 , which, in turn, shall form a sequence (๐ ๐ ) ๐โฅ0 of natural numbers. The initial first five terms for ๐ = 0,1,3,4,5 of (๐ ๐ ) ๐โฅ0 are
1, 3, 299, 28349043, 21262618727925419 . Table 2.
Number of fuzzy matrices ๐ ๐ .1235222925834942417851639819 .37095516161844674407648 .213125629499534497 .66871947673366 .33554432255 7925419212626187265536164 2834904351293 2991642 3211 1100 )()002416(2)000290(000027( OEISin availableNotfOEISinAOEISinAnOEISinAn nn
4. Conclusions and Further Research
The investigate about the classification of fuzzy matrices is an important aspect of fuzzy matrix theory. In this paper, we have classified and counted all distinct fuzzy matrices of order ๐ by incorporating Muraliโs equivalence relation ([24], [33], [35]). For each ๐ โ โ , listing the number of all distinct the fuzzy matrices of order ๐ will take a considerable amount of time due to enormous and the quick exponential growth of the number ๐ ๐ . It will be a very complex task to calculate many more terms of the sequence (๐ ๐ ) using our counting technique. The classification of the counting problem can successfully be extended to some other special classes of fuzzy matrices. The study of fuzzy matrices can be effectively used in matrix theory, combinatorics as well as in geometry. It might be interesting to attack the classification of counting problem from different perspectives with more efficiently. This will surely indicate the way to further research. There are two open problems according to this area of classification are as follows: Problem 4.1.
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