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### An investigation on the $n$-fold IVRL-filters in triangle algebras

Mathematica Bohemica

The present study aimed to introduce $n$-fold interval valued residuated lattice (IVRL for short) filters in triangle algebras. Initially, the notions of $n$-fold (positive) implicative IVRL-extended filters and $n$-fold (positive) implicative triangle algebras were defined. Afterwards, several characterizations of the algebras were presented, and the correlations between the $n$-fold IVRL-extended filters, $n$-fold (positive) implicative algebras, and the Gödel triangle algebra were discussed.

### Characterizing matrices with $𝐗$-simple image eigenspace in max-min semiring

Kybernetika

A matrix $A$ is said to have $𝐗$-simple image eigenspace if any eigenvector $x$ belonging to the interval $𝐗=\left\{x:\underline{x}\le x\le \overline{x}\right\}$ is the unique solution of the system $A\otimes y=x$ in $𝐗$. The main result of this paper is a combinatorial characterization of such matrices in the linear algebra over max-min (fuzzy) semiring. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability) that...

### Computing the greatest $𝐗$-eigenvector of a matrix in max-min algebra

Kybernetika

A vector $x$ is said to be an eigenvector of a square max-min matrix $A$ if $A\otimes x=x$. An eigenvector $x$ of $A$ is called the greatest $𝐗$-eigenvector of $A$ if $x\in 𝐗=\left\{x;\underline{x}\le x\le \overline{x}\right\}$ and $y\le x$ for each eigenvector $y\in 𝐗$. A max-min matrix $A$ is called strongly $𝐗$-robust if the orbit $x,A\otimes x,{A}^{2}\otimes x,\cdots$ reaches the greatest $𝐗$-eigenvector with any starting vector of $𝐗$. We suggest an $O\left({n}^{3}\right)$ algorithm for computing the greatest $𝐗$-eigenvector of $A$ and study the strong $𝐗$-robustness. The necessary and sufficient conditions for strong $𝐗$-robustness are introduced and an efficient...

### Congruences and homomorphisms on $\Omega$-algebras

Kybernetika

The topic of the paper are $\Omega$-algebras, where $\Omega$ is a complete lattice. In this research we deal with congruences and homomorphisms. An $\Omega$-algebra is a classical algebra which is not assumed to satisfy particular identities and it is equipped with an $\Omega$-valued equality instead of the ordinary one. Identities are satisfied as lattice theoretic formulas. We introduce $\Omega$-valued congruences, corresponding quotient $\Omega$-algebras and $\Omega$-homomorphisms and we investigate connections among these notions. We prove...

### Controllable and tolerable generalized eigenvectors of interval max-plus matrices

Kybernetika

By max-plus algebra we mean the set of reals $ℝ$ equipped with the operations $a\oplus b=max\left\{a,b\right\}$ and $a\otimes b=a+b$ for $a,b\in ℝ.$ A vector $x$ is said to be a generalized eigenvector of max-plus matrices $A,B\in ℝ\left(m,n\right)$ if $A\otimes x=\lambda \otimes B\otimes x$ for some $\lambda \in ℝ$. The investigation of properties of generalized eigenvectors is important for the applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval) entries...

### Eigenspace of a circulant max–min matrix

Kybernetika

The eigenproblem of a circulant matrix in max-min algebra is investigated. Complete characterization of the eigenspace structure of a circulant matrix is given by describing all possible types of eigenvectors in detail.

### Extension of $n$-dimensional Euclidean vector space ${E}^{n}$ over $ℝ$ to pseudo-fuzzy vector space over ${F}_{p}^{1}\left(1\right)$.

International Journal of Mathematics and Mathematical Sciences

### Foldness of Commutative Ideals in BCK-algebras

Discussiones Mathematicae - General Algebra and Applications

This paper deals with some properties of n-fold commutative ideals and n-fold weak commutative ideals in BCK-algebras. Afterwards, we construct some algorithms for studying foldness theory of commutative ideals in BCK-algebras.

### Fuzzy data in statistics

Kybernetika

The development of effective methods of data processing belongs to important challenges of modern applied mathematics and theoretical information science. If the natural uncertainty of the data means their vagueness, then the theory of fuzzy quantities offers relatively strong tools for their treatment. These tools differ from the statistical methods and this difference is not only justifiable but also admissible. This relatively brief paper aims to summarize the main fuzzy approaches to vague data...

### Fuzzy Euclidean ideals.

APPS. Applied Sciences

### Fuzzy ideals extensions of ordered semigroups.

Lobachevskii Journal of Mathematics

### Fuzzy structures of PI($\ll ,\subseteq ,\subseteq$) BCK-ideals in hyper BCK-algebras.

International Journal of Mathematics and Mathematical Sciences

### Fuzzy subcoalgebras and duality.

Bulletin of the Malaysian Mathematical Sciences Society. Second Series

### Generalized fuzzy $k$-ideals of semirings with interval-valued membership functions.

Bulletin of the Malaysian Mathematical Sciences Society. Second Series

### Generated fuzzy implications and fuzzy preference structures

Kybernetika

The notion of a construction of a fuzzy preference structures is introduced. The properties of a certain class of generated fuzzy implications are studied. The main topic in this paper is investigation of the construction of the monotone generator triplet $\left(p,i,j\right)$, which is the producer of fuzzy preference structures. Some properties of mentioned monotone generator triplet are investigated.

### Generating methods for principal topologies on bounded lattices

Kybernetika

In this paper, some generating methods for principal topology are introduced by means of some logical operators such as uninorms and triangular norms and their properties are investigated. Defining a pre-order obtained from the closure operator, the properties of the pre-order are studied.

### Homotopy method for solving fuzzy nonlinear equations.

APPS. Applied Sciences

### Maximal column rank preservers of fuzzy matrices

Discussiones Mathematicae - General Algebra and Applications

This paper concerns two notions of rank of fuzzy matrices: maximal column rank and column rank. We investigate the difference of them. We also characterize the linear operators which preserve the maximal column rank of fuzzy matrices. That is, a linear operator T preserves maximal column rank if and only if it has the form T(X) = UXV with some invertible fuzzy matrices U and V.

### Maximal solutions of two–sided linear systems in max–min algebra

Kybernetika

Max-min algebra and its various aspects have been intensively studied by many authors [1, 4] because of its applicability to various areas, such as fuzzy system, knowledge management and others. Binary operations of addition and multiplication of real numbers used in classical linear algebra are replaced in max-min algebra by operations of maximum and minimum. We consider two-sided systems of max-min linear equations $A\otimes x=B\otimes x$, with given coefficient matrices $A$ and $B$. We present a polynomial method for...

### Monotone interval eigenproblem in max–min algebra

Kybernetika

The interval eigenproblem in max-min algebra is studied. A classification of interval eigenvectors is introduced and six types of interval eigenvectors are described. Characterization of all six types is given for the case of strictly increasing eigenvectors and Hasse diagram of relations between the types is presented.

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