This paper is dealing with solvability of interval systems of linear equations in max-min algebra. Max-min algebra is the algebraic structure in which classical addition and multiplication are replaced by ⊕ and \kr, where a⊕b=max{a,b},a\krb=min{a,b}. The notation \mbfA\krx=\mbfb represents an interval system of linear equations, where \mbfA=[\pA,\nA] and \mbfb=[\pb,\nb] are given interval matrix and interval vector, respectively. We can define several types of solvability of interval systems. In this paper, we define the T4 and T5 solvability and give necessary and sufficient conditions for them.
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.
Max-min algebra is an algebraic structure in which classical addition and multiplication are replaced by ⊕ and \kr, where a⊕b=max{a,b},a\krb=min{a,b}. The notation \mbfA\kr\mbfx=\mbfb represents an interval system of linear equations, where \mbfA=[\pA,\nA], \mbfb=[\pb,\nb] are given interval matrix and interval vector, respectively, and a solution is from a given interval vector \mbfx=[\px,\nx]. We define six types of solvability of max-min interval systems with bounded solution and give necessary and sufficient conditions for them.
Max-min algebra and its various aspects have been intensively studied by many authors \cite{Baccelli,Green79} 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
\begin{math}\emph{A}\otimes\emph{x}= B\otimes\emph{x}\end{math}, with given coefficient matrices \emph{A} and \emph{B}. We present a polynomial method for finding maximal solutions to such systems, and also when only solutions with prescribed lower and upper bounds are sought.
In max-min algebra the standard pair of operations plus and times is replaced by the pair of operations maximum and minimum, respectively. A max-min matrix A is called strongly robust if the orbit x,A⊗x,A2⊗x,… reaches the greatest eigenvector with any starting vector. We study a special type of the strong robustness called the strong \textit{\textbf{X}}-robustness, the case that a starting vector is limited by a lower bound vector and an upper bound vector. The equivalent condition for the strong \textit{\textbf{X}}-robustness is introduced and efficient algorithms for verifying the strong \textit{\textbf{X}}-robustness is described. The strong \textit{\textbf{X}}-robustness of a max-min matrix is extended to interval vectors \textit{\textbf{X}} and interval matrices \textit{\textbf{A}} using for-all-exists quantification of their interval and matrix entries. A complete characterization of AE/EA strong \textit{\textbf{X}}-robustness of interval circulant matrices is presented.
A matrix A is said to have \mbox{\boldmathX}-simple image eigenspace if any eigenvector x belonging to the interval \boldmathX={x:x−−≤x≤x¯¯¯} containing a constant vector is the unique solution of the system A⊗y=x in \mbox{\boldmathX}. The main result of this paper is an extension of \mbox{\boldmathX}-simplicity to interval max-min matrix \boldmathA={A:A−−≤A≤A¯¯¯¯} distinguishing two possibilities, that at least one matrix or all matrices from a given interval have \mbox{\boldmathX}-simple image eigenspace. \mbox{\boldmathX}-simplicity of interval matrices in max-min algebra are studied and equivalent conditions for interval matrices which have \mbox{\boldmathX}-simple image eigenspace are presented. 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 have been studied in max-min and max-plus algebras.