Maximum solution of fuzzy relation equation

P = {p,:i ϵ N_n}, Q = {q_j, : j ϵ N_m}, R = {r_k : k ϵ N_k} are considered as three referential sets and A ϵ R (P, Q;I), X ϵ R (Q, R; I), B ϵ R (P, R; I) are three fuzzy relations. We assume that these fuzzy relations are constrained to each other such that A o X = B. If two of the three components are known and other is to be calculated, then it is called fuzzy relation equation. When A and X are known, the problem is very easy to be solved by direct application of max-min composition. The problem is not easily solvable when one matrix in the left side is not known. The inverse of the matrices cannot be applied, because matrices may not be invertible. In that case solution may not exist. Again if solution exists, it may not be unique. In fuzzy relation equation A o X = B. we consider X is unknown. This equation can be partitioned into a set of fuzzy relation equations of the form A°X(:, i) = B(:, i), ∀i For convenience we shall write A o x = b, where A ϵ M_mn (I) is an m×n fuzzy matrix and B ϵ I^m is a fuzzy m-vector and x ϵ Iⁿ is a fuzzy n-vector. Here "o" max-T composition, where T is t-norm. Our purpose is to determine x ϵ Iⁿ satisfying A ° x = b. In this paper we consider the t-norm "Minprod". A theoretical aspect of the existence of the solution is discussed. It is theoretically established that one solution is maximum. After different Definition and Theorems MATLAB function scripts are constructed. Examples with input and output are kept in an appendix.