Abstract
The index matrix (IM) is an extension of the ordinary matrix with indexed rows and columns. Over IMs’ standard matrix operations are defined and a lot of other ones that do not exist in the standard case. Intuitionistic fuzzy IMs (IFIMs) are modification of the IMs, when their elements are intuitionistic fuzzy pairs (IFPs). Extended IFIMs are IFIMs whose indices of the rows and columns are evaluated by IFPs. Different operations, relations and operators over IFIMs, and some specific ones, are defined for EIFIMs. In the paper, twelve new level operators are defined for EIFIMs and in the partial case, over IFIMs. The proposed level operators fall into two groups: operators that change the values of the EIFIM elements and operators that change the IFPs associated to the indices of the rows and columns. The basic properties of the operators are studied.
MSC:
03E72; 11C20
1. Introduction
The matrix theory is one of the basic areas of algebra (see, e.g., [,]). Different operations are defined over matrices, but each one of these operations is possible only when some conditions hold. For example, we cannot sum two matrices with different dimensions.
The concept of an index matrix (IM) was introduced in 1987 in [] to enable two matrices with different dimensions to be summed. Later, this concept was extended essentially and not only operations, but also relations and operators were defined over IM. Their properties were discussed in a series of papers and a comprehensive overview can be found in the book [].
Initially, by analogy with the standard matrices, the elements of the IMs were real numbers. Later, these elements obtained the form of sentences and predicates, functions, etc. One of the basic IM-modifications are intuitionistic fuzzy IMs (IFIMs) whose elements are intuitionistic fuzzy pairs (IFPs) []. A further modification are the extended IFIMs (EIFIMs), i.e., IFIMs whose indices of the rows and columns are evaluated by IFPs.
The 3-dimensional intuitionistic fuzzy index matrices (3-D IFIMS) introduced in [] were further studied and new definitions were proposed in []. Subsequently, some basic operations and modal operators over the 3-D IFIMS were investigated in [,]. Some applications of the apparatus of IM have been discussed in [,]. Other examples of applications to number theory are given in [].
To widen the area of application of IM, it is necessary to enrich the theory with additional operators. Following this, in the present study we introduce new level operators for better treatment and aggregation of information. We also establish some of the level operators properties.
In [], for the first time, level operators were defined over IMs for the cases when the elements are real numbers in a fixed (finite) interval. Here, we continue this research, expanding it to the case with IFIMs having as elements IFPs, keeping all notations previously used to make the differences more pronounced.
2. Preliminaries
Initially, we give some remarks on IFPs. The IFP is an object with the form , where and , that is used as a tool for evaluation. Its components (a and b) are interpreted as degrees of membership and non-membership, or degrees of validity and non-validity, or degree of correctness and non-correctness, etc.
Let us have two IFPs and .
The following relations have been defined in []:
We define analogous of operations “conjunction” and “disjunction”:
When “∘” is one of these operations, then we see that it has a pair of two sub-operations. For example, if “∘” is “∧”, then its sub-operations are “min” and “max”, while if “∘” is “∨”, then its sub-operations are “max” and “min”. Let us denote these operations by “” and “”, respectively.
In [], definitions of 185 operations’ “implication” are given, while in [,,] for each one of these implications, three conjunctions and three disjunctions are juxtaposed.
Let be a fixed set. By IFIM with index sets K and L , we denote the object:
where for every :
For brevity, we can denote the above object by , where
for and :
Now, for the mentioned above sets K and L, the EIFIM is defined by:
where for every :
and here and below,
Therefore,
Let for :
Then
For the EIFIMs , operations that are analogous to the standard matrix operations of addition and multiplication, as well as other specific ones (see [,]) are defined as given below. Some technical oversights existing in [] were pointed out in [].
Addition-
where
and
Termwise multiplication-(∘)
where
for
for
and
Let be another operation and let it have sub-operations .
Multiplication-()
where
and
For example, when and , we obtain operation multiplication-() from [].
Let
and
Then
and
Let
then
A lot of relations are defined over two EIFIMs. Here, we use only five of them:
The strict relation “inclusion about value”:
The non-strict relation “inclusion about value”:
The strict relation “inclusion about dimension”:
The non-strict relation “inclusion about dimension”:
The relation “equality about dimension”:
3. Definition and Properties of the New Level Operators
The proposed new level operators fall into two groups. The operators from the first group are applicable as over EIFIMs, as well as over IFIMs, while the operators from the second group are specific only for EIFIMs.
3.1. Definition and Properties of the Operators from the First Group
Let us have the IM , where is an IFP. Then, for two fixed numbers such that we define:
where
where
where
where
Let for :
denote the zero and unit EIFIMs with index sets and . Obviously, the following equalities are valid for every two fixed numbers such that :
Theorem 1.
For each EIFIM A and for every two IFPs :
Proof.
Let the above EIFIM A be given. Then
where
and
If , then
if , then
but ; if , then
and hence, ; if , then
and hence, .
Therefore,
i.e.,
The remaining equalities are proved in the same manner. □
Corollary 1.
For each EIFIM A and for every two IFPs :
Theorem 2.
Let the two EIFIMs A and B be given and let be an arbitrary IFP. Then
- (a)
- ,
- (b)
- ,
- (c)
- ,
- (d)
- ,
- (e)
- ,
- (f)
- ,
- (g)
- ,
- (h)
- ,
- (i)
- ,
- (j)
- ,
- (k)
- ,
- (l)
- ,
- (m)
- ,
- (n)
- ,
- (o)
- ,
- (p)
- .
Proof.
(p) Let EIFIM A have the above form and let
where is an IFP.
From definition of operation , we have:
where
and
Sequentially, we will study the different cases for .
1. If , then
and therefore, the elements with indices will coincide in the two IMs and .
2. If , then
and therefore, the elements with indices will coincide in the two IMs and .
3. If , then
Now, there are the following four subcases:
3.1. if
and
then these elements will keep their values in and in , respectively and their conjunction will coincide with the value in EIFIM .
3.2. if
and
i.e.,
then only the a-element will keep its value in , while the b-element will obtain value in . Therefore, the value of in EIFIM will be equal to its corresponding value in the EIFIM .
3.3. if
and
i.e.,
then only the b-element will keep its value in , while the a-element will obtain value in . Therefore, the value of in EIFIM will be equal to its corresponding value in IM .
3.4. if
and
then the a- and b-elements will obvain value in IMs and . Therefore, the value of in IM will be equal to and to its corresponding value in IM .
4. If , then and therefore, the elements with indices will coincide in the two IMs and .
Hence, relation = between the two IMs exists.
The other cases are proved in the same manner. □
If in Theorem 2 we replace operation by operation or , the inequalities will keep their forms and the proofs will be analogous.
Now, we will modify the operator automatic reduction from [] to the form:
where are index sets with the following property:
Therefore, this operator removes the rows and columns of EIFIM A that contain only elements .
3.2. Definition and Properties of the Operators from the Second Group
Let us have the EIFIM
and for the two numbers such that :
where
and
where
and
where
and
where
and
where
and
where
and
where
and
where
and
Theorem 3.
For each EIFIM A and for every two IFPs :
where .
We omit the proof of Theorem 3 and the next theorem (Theorem 4) as they may be proved similarly to Theorem 2.
Finally, we modify the operator automatic reduction from Section 3.1 to the forms:
where is the index sets with the following property:
where is the index sets with the following property:
Therefore, this operator removes the rows and columns of EIFIM A in which indices have evaluation .
Theorem 4.
For each EIFIM A and for every two IFPs :
Theorem 1 keeps its sense when we change operator with each one of the operators and , where .
Theorem 2 keeps also its sense for each one of operations when we change operator with each one of the operators and , where , and relations and with relations and , respectively.
4. An Example
A possible application of the newly proposed level operators to assessment of the student’s training is considered.
Let us have s students that have (by the current moment) scores for Let these students study d different disciplines , that have scores received by the previous procedures for
The student’s scores (following previous examinations) after each examination are calculated by the formula
Obviously, is an IFP.
The discipline’s score (after m previous training procedures) after each examination of the students , is calculated by the formula
where is the number of the students who received high score, and is the number of the students who had poor score. Therefore, the number corresponds to the number of the students who had not attended the examination and obviously, is an IFP.
The sets of triples
and
are the indices of the EIFIM
and IFP corresponds to the evaluation of the ith student on the examination of jth discipline. The scores as IFPs can be calculated in the following manner: , where r is the number of all problems that the student had to solve, p is the number of correctly solved problems, q is the number of wrongly solved problems and is the number of the problems that were not solved, e.g., for the lack of time.
Using operators and over EIFIM A, we will determine the higher evaluations, using operators and , the lower evaluations, using operators and , the students with higher scores, using operators and , the students with lower scores, using operators and , the disciplines with higher scores, using operators and , the disciplines with lower scores.
It is possible to describe the elements in the matrix A, i.e., the scores, as natural or real numbers, instead of IFPs, when this is convenient.
5. Conclusions
The IMs were introduced to resolve some particular problems related to the extensions of some modifications of Petri nets []. Over the years, they have been enriched with different relations, operations and operators. In addition to the standard two dimensional IMs, 3- and n- dimensional IMs were defined. With [] and the current research, we initiate a new direction of research over IM—introduction and studying of group of new level operators. It is important to note that many of the operations, relations and operators do not exist in the ordinary matrices. Moreover, the new level operators are different in sense compared to the fuzzy and intuitionistic fuzzy types of level operators (cf. []).
In the near future, other properties of the introduced level operators will be studied. These operators can find applications in different areas, such as intuitionistic fuzzy graphs, intuitionistic fuzzy cognitive maps and others (see []), that will be deeply and thoroughly investigated and analyzed in sufficient detail in a series of future studies.
Author Contributions
Investigation, K.A., P.V. and O.R.; Writing—original draft, K.A., P.V. and O.R.; Writing—review and editing, K.A., P.V. and O.R. All authors have read and agreed to the published version of the manuscript.
Funding
This research was funded by Bulgarian National Science Fund, grant number KP-06-N22/1/2018 “Theoretical research and applications of InterCriteria Analysis”.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
No new data were created or analyzed in this study. Data sharing is not applicable to this article.
Conflicts of Interest
The authors declare no conflict of interest.
References
- Brown, W.C. Matrices and Vector Spaces; Marcel Dekker: New York, NY, USA, 1991. [Google Scholar]
- Lankaster, P. Theory of Matrices; Academic Press: New York, NY, USA, 1969. [Google Scholar]
- Atanassov, K. Generalized index matrices. Comptes Rendus De L’Academie Bulg. Des Sci. 1987, 40, 15–18. [Google Scholar]
- Atanassov, K. Index Matrices: Towards an Augmented Matrix Calculus; Springer: Cham, Switzerland, 2014. [Google Scholar]
- Atanassov, K. Intuitionistic Fuzzy Logics; Springer: Cham, Switzerland, 2017. [Google Scholar]
- Traneva, V. On 3-dimensional intuitionistic fuzzy index matrices. Notes Intuit. Fuzzy Sets 2014, 20, 59–64. [Google Scholar]
- Traneva, V. Internal operations over 3-dimensional extended index matrices. Proc. Jangjeon Math. Soc. 2015, 18, 547–569. [Google Scholar]
- Traneva, V. More basic operations and modal operators over 3-dimensional intuitionistic fuzzy index matrices. Notes Intuit. Fuzzy Sets 2014, 20, 17–25. [Google Scholar]
- Parvathi, R.; Thilagavathi, S.; Thamizhendhi, G.; Karunambigai, M.G. Index matrix representation of intuitionistic fuzzy graphs. Notes Intuit. Fuzzy Sets 2014, 20, 100–108. [Google Scholar]
- Traneva, V. One application of the index matrices for a solution of a transportation problem. Adv. Stud. Contemp. Math. 2016, 26, 703–715. [Google Scholar]
- Leyendekkers, J.; Shannon, A.; Rybak, J. Pattern Recognition: Modular Rings & Integer Structure; Raffles KvB Monograph No. 9; Raffles KvB Institute: North Sydney, Australia, 2007. [Google Scholar]
- Atanassov, K. Level operators over index matrices. Part 1: Index matrices with elements from a fixed interval. Annu. Inform. Sect. Union Sci. Bulg. 2019/2020, 10, 1–12. [Google Scholar]
- Angelova, N.; Stoenchev, M. Intuitionistic fuzzy conjunctions and disjunctions from first type. Annu. Inform. Sect. Union Sci. Bulg. 2015/2016, 8, 1–17. [Google Scholar]
- Angelova, N.; Stoenchev, M.; Todorov, V. Intuitionistic fuzzy conjunctions and disjunctions from second type. Issues Intuit. Fuzzy Sets Gen. Nets 2017, 13, 143–170. [Google Scholar]
- Angelova, N.; Stoenchev, M. Intuitionistic fuzzy conjunctions and disjunctions from third type. Notes Intuit. Fuzzy Sets 2017, 23, 29–41. [Google Scholar]
- Atanassov, K. On Intuitionistic Fuzzy Sets Theory; Springer: Berlin, Germany, 2012. [Google Scholar]
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