1. Introduction
The eigenproblem for a fuzzy matrix corresponds to finding a stable state (or all stable states) of the complex discrete-events system described by the given transition matrix and fuzzy state vectors. Therefore, the investigation of the eigenspace structure in fuzzy algebras is important for application. This problem has been solved in several types of so-called extremal algebras.
A max-T fuzzy algebra is defined over the interval
and uses, instead of the conventional operations of addition and multiplication, the operations of maximum and one of the triangular norms, the so-called t-norm. These operations are extended in a natural way to the Cartesian products of vectors and matrices. The t-norms together with the t-conorms play a key role in fuzzy theory. These functions have applications in many areas, such as decision making, statistics, game theory, information and data fusion, probability theory, and risk management.
Although there exist various t-norms and families of t-norms (see, e.g., [1] ), let us mention the several main t-norms: the Łukasiewicz t-norm, the Gödel t-norm, the nilpotent minimum t-norm, the product t-norm, and the drastic t-norm.
The Łukasiewicz t-norm is computed as
The Gödel t-norm is the simplest t-norm and the conjunction is defined as the minimum of the entries, i.e.,
The nilpotent minimum t-norm is defined by
The definition of the product t-norm is
The drastic triangular t-norm is “the weakest norm” and the basic example of a non-divisible t-norm on any partially ordered set. The drastic triangular t-norm is defined as follows:
Recently, Gavalec et al. [2] [3] investigated the steady states of max-Łukasiewicz fuzzy systems and monotone interval eigenproblem in max-min algebra, Rashid et al. [4] discussed the eigenspace structure of a max-product fuzzy matrix and Gavalec et al. [5] studied the eigenspace structure of a max-drast fuzzy matrix. In this paper, based on these works, we further study eigenproblem. We investigate the eigenvectors in a max-T algebra, study monotone eigenvectors in max-nilpotent-min algebra, discuss the relation between the monotone eigenvectors in max-T algebra and max-drast algebra, and illustrate the relations among eigenspaces in these algebras by some examples.
2. Eigenvectors in a Max-T Algebra
Let T be one of the triangular norms used in fuzzy theory, let us denote the real unit interval
by I. By the max-T algebra we understand the triple
with the binary operations
and
on I. For given natural
, we write
. The set of all permutations on N will be denoted by
. The notations
and
denote the set of all vectors and all square matrices of a given dimension
over I, respectively. The operations
and
are extended to matrices and vectors in the standard way.
The eigenproblem for a given matrix
in max-T algebra consists in finding an eigenvector
for which
holds true. The eigenspace of
is denoted by
Theorem 2.1. Let
be three triangular norms on I,
and
. If
,
and
, then
.
Proof. If
and
, then
When
, we have that
i.e.,
. Thus,
.
The theorem is proved.
The investigation of the eigenspace structure can be simplified by permuting any vector
to a non-decreasing form.
For given permutations
, we denote by
the matrix with rows permuted by
and columns permuted by
, and we denote by
the vector permuted by
.
Theorem 2.2. (Gavalec [6] ). Let
,
and
. Then
if and only if
.
We say a vector
is increasing if
holds for any
and strictly increasing if
whenever
. The set of all increasing eigenvectors of a matrix A is denoted by
and the set of all strictly increasing eigenvectors of a matrix A is denoted by
. Similar notation
and
will be used without the condition
.
Theorem 2.3. Let
and
. Then
if and only if for every
the following hold.
Proof. By definition,
is equivalent with the condition
which is equivalent to
for every
and
for some
.
The theorem is proved.
Theorem 2.4. Let
and
. If
, then
1)
for all
,
2)
.
Proof. If
, then it follows from Theorem 2.3 that
When
,
, this is a contradiction. Thus,
for every
. Noting that
is the largest triangular normon I, we see that
and hence
.
The theorem is proved.
In particular, if
with
, then
3. Eigenvectors in Max-Łukasiewicz Algebra
The following theorem contains several logical consequences of the definition of Łukasiewicz triangular norm.
Theorem 3.1. (Rashid et al. [7] ). Let
. Then
1)
if and only if
or
,
2)
if and only if
or (
and
),
3)
if and only if
,
4)
if and only if
,
5) if
, then
.
Combining Theorem 2.3 with Theorem 3.1, we have the following theorem.
Theorem 3.2. (Rashid et al. [7] ). Let
and
. Then
if and only if for every
the following hold:
1)
for every
and
,
2) if
, then
or
for some
,
3) if
, then
for some
.
The following theorem describes necessary conditions under which a given square matrix can have a strictly increasing eigenvector.
Theorem 3.3. (Rashid et al. [7] ). Let
. If
, then the following conditions are satisfied
1)
for all
and
,
2)
.
The following theorem describes necessary and sufficient conditions under which a three-dimensional fuzzy matrix has a strictly increasing eigenvector.
Theorem 3.4. (Rashid et al. [7] ). Let
. Then
if and only if the following conditions are satisfied
1)
, for all
and
,
2)
, or
,
3)
.
Example 3.1. Let us consider the matrix
Matrix A satisfies conditions (1)-(3) in Theorem 3.4, hence
and
4. Eigenvectors in Max-Min Algebra
In the case of the max-min (called also: bottleneck) algebras, the eigenproblem has been studied by many authors and interesting results describing the structure of the eigenspace have been found (see [3] [8] [9] [10] [11] [12] ). In particular, algorithms have been suggested for computing the maximal eigenvector of a given max-min matrix (see [13] ).
If the binary operation
coincides with the minimum operation, then the strictly increasing eigenspace
can be described as an interval of strictly increasing eigenvectors, where the bounds
of the interval are defined as follows
If a maximum of an empty set should be computed in the above definition of
, then we use the fact that
by usual definition.
The following theorem has been proved in [6] .
Theorem 4.1. (Gavalec [6] ). Let
and
be a strictly increasing vector. Then
if and only if
, i.e.,
Hence, in view of Theorem 4.1, the structure of
has been completely described for any
.
5. Eigenvectors in Max-Nilpotent-Min Algebra
We know that the nilpotent minimum norm
is left-continuous and the R-implication generated from
is defined by
Moreover, it follows from Proposition 2.5.2 in [14] that
and
form an adjoint pair, i.e., they satisfy the following residual principle
and
If
, then
is equivalent with the two conditions:
1) for any
,
,
2) there exist
such that
.
For
,
1) if
, then
;
2) if
, then
if and only if
a) when
,
,
,
b) when
,
and
i.e.,
and hence
;
3) if
, then
and
6. Eigenvectors in Max-Product Algebra
For every vectors
, define the quotient vector
by
Then,
if and only if
fulfills the following inequalities
Noting that for any
,
Thus, it follows from Theorem 2.3 that
Theorem 6.1. (Rashid et al. [4] ). Suppose that
and
. Then
if and only if for every
the following two conditions hold.
1)
for every
,
2)
or
for some
.
When
,
and it follows from the proof of Theorem 2.4 that
, i.e.,
.
Thus, the following theorem is a corollary of Theorem 2.4.
Theorem 6.2. (Rashid et al. [4] ). If
and
, then the following conditions satisfied
1)
for all
,
2)
.
This Theorem describes necessary conditions which a square matrix can have an increasing eigenvector.
7. Eigenvectors in Max-Drast Algebra
For
, we have
for every
.
Theorem 7.1. (Gavalec et al. [5] ). Let
and
. Then
if and only if for every
the following conditions hold
1)
for every
,
2) if
, then
,
3) for some
,
.
Moreover, the following theorem describes necessary and sufficient conditions which a square matrix possesses a strictly increasing eigenvector.
Theorem 7.2. (Gavalec et al. [5] ). Let
. Then
if and only if the following conditions are satisfied
1)
for all
,
2)
for all
with
,
3)
for all
with
,
4)
.
When
and
i.e.,
. This shows that conditions (1) and (4) in Theorem 7.2 are also straightforward consequences of Theorem 2.4.
The next theorem characterizes all the eigenvectors of a given matrix. In other words, the theorem completely describes the eigenspace structure.
Theorem 7.3. (Gavalec et al. [5] ). Let
,
, and
. Then
if and only if the following conditions are satisfied
1)
for all
with
,
2) if
, then
for all
with
,
3) if
, then
,
4) if
for some
, then
.
8. The Relations among These Eigenspaces
Now we discuss the relation between the monotone eigenvectors in max-T algebra and max-drast algebra.
Theorem 8.1. Let
. If
and
, then
Proof. Assume that
. For each
, when
, it follows from Theorem 2.4 that
. If
, then
Thus,
.
When
,
i.e., there exist some
such that
Therefore,
by Proposition 3.3 in [5] .
The theorem is proved.
Finally, we illustrate the relations among eigenspaces in these algebras by two examples.
Example 8.1. Let
Then
Thus,
, but
. This illustrates that the condition
is necessary in Theorem 6.2. Moreover, by a simple computation, we see that
i.e.,
.
This example shows that
even if
.
Example 8.2. Let
Then the conditions (1)-(4) hold in Theorem 3.4 in [5] . Thus,
. But,
i.e.,
and
.
9. Conclusions and Further Works
The eigenproblem for a fuzzy matrix corresponds to finding a stable state of the complex discrete-events system described by the given transition matrix and fuzzy state vectors and the investigation of the eigenspace structure in fuzzy algebras is important for application. Gavalec et al. [2] [3] have investigated the steady states of max-Łukasiewicz fuzzy systems, Rashid et al. [4] and Gavalec et al. [5] have discussed the eigenspace structure of a max-product fuzzy matrix and a max-drast fuzzy matrix, respectively.
In this paper, we investigated the eigenvectors in a max-T algebra, discussed monotone eigenvectors in max-nilpotent-min algebra, and studied the relation between the monotone eigenvectors in max-T algebra and max-drast algebra.
In a forthcoming paper, we will further investigate monotone eigenvectors in max-nilpotent-min algebra and max-T algebra.
Acknowledgements
This work is funded by College Students Practice Innovation Training Program (201610324027Y).