The systematic structure of finite semigroups allows analysis of finite semigroups in terms of inner algebraic structure and behavior. The paper will talk about the decomposition method used on the basis of the relations of Green, ideal theory and idempotent elements to classify the semigroup elements. It concentrates on Rees theorem and its expression in a matrix form of entirely simple semigroups, 0-simple semigroups, which makes it possible to canonically describe the structure. Furthermore, algorithmic techniques are designed to locate ideals, which computes the equivalence classes, an efficient Rees decomposition and factorization. The augmented computational tractability and scaling of large semigroups can be achieved by using the proposed procedures. The experimental analysis shows that the decomposition accuracy and simplification of computations increase. The outcomes are the contribution to the field of the algebraic theory and the use in the automata, cryptography and the discrete optimization systems to ensure high level of structure insight and efficiency in implementation.