Some Studies On Matrix Equations In ∧-â†' Composition Over Complete Brouwer Lattices

In this paper, we will investigate the matrix equation in composition over a complete Brouwerian lattice. First, we give an equivalent condition for the solvability of this equation, i.e. theorem 2.2.3 in this paper: The matrix equation in composition over a complete Brouwerian lattice is solvable iff is the minimal solution of this equation. In order to find the equivalent condition, we start from some simple forms of the equation , first give equivalent conditions for the solvability of these equations, and then consider the complex form. In order to determine the solution set of the equation , by the means of meet-irreducible element and irredundant finite meet-decomposition, we first obtain the maximal solutions to the simple equation in the case that b has an irredundant finite meet-decomposition, and then consider the relation between the equation and the equation , based on this, we obtain the maximal solutions to the equation in the case that each element of the matrix B has an irredundant finite meet-decomposition and so determine its solution set completely. Secondly, for a more general matrix equation in composition, we obtain another equivalent condition for the solvability of this equation and give an equivalent characterization to the solution set of the equation.Finally, in order to show some application of this equation, we consider the perturbation of the equation and obtain some upper perturbed elements and their perturbed interval to this equation.