Identify Solutions to Systems of Linear Latin for Square Equations over Maxmin-ω

Maxmin-\omega algebra is a mathematical system that generalizes maxmin algebra by introducing the parameter \omega (0 \omega \leq 1), which regulates the algebraic operations to enhance its applicability in optimization and decision-making processes. When \omega=1, the system corresponds to the max operation, whereas for \omega approaching 0, it behaves like the min operation. This research investigates the solution characteristics of a linear equation system in maxmin-\omega algebra, specifically A \otimes_{\omega} \textbf{x} = \textbf{b}, where A is a Latin square matrix. Understanding these solutions is crucial for determining the conditions of existence and uniqueness, which will ultimately influence the development of more efficient solution methods for various applications. Furthermore, the study analyzes the impact of the value of \omega and the matrix permutation structure on the solutions of the system. This study employs an analytical approach utilizing maxmin-\omega algebra theory to determine solution existence and assess the impact of \omega variations in linear equations with Latin square matrices. The results reveal that the solution existence heavily depends on the composition of matrix A and the vector \textbf{b}. We show that in specific cases where the matrix \( A \) is a Latin square and the vector \( \mathbf{b} \) satisfies certain constraints, the system has a unique solution in both the max-plus (\(\omega = 1\)) and min-plus (\(\omega = \frac{1}{n}\)) approaches. Moreover, column permutations of A do not affect the existence of solutions. However, row and element permutations alter the system structure, meaning solutions are not always guaranteed.