Interval Fuzzy Matrix Equations

This paper is dealing with solvability of interval matrix equations in fuzzy algebra. Fuzzy algebra is an algebraic structure, in which classical addition is replaced by maximum and the second operation, multiplication, is replaced by minimum. Fuzzy equations have found a broad area of applications in causal models which emphasize relationships between input and output variables. They are used in diagnosis models or models of non-deterministic systems. We shall deal with the both-sided matrix equations. If we replace elements in matrices with intervals of possible values, we obtain an interval matrix equation. We can define several types of solvability of interval matrix equations. In this paper, we shall deal with four of them. We define the tolerance, weak tolerance, left-weak tolerance, and right-weak tolerance solvability and provide polynomial algorithms for checking of them. We prove, that in fuzzy algebra, weak tolerance, left-weak tolerance, and right-weak tolerance solvability are equivalent to each other.