Several problems on subsystems

Introduces the concept of subsystems closure and studies their existence, uniqueness and structure. Based on this concept, introduces the concept of finitely generated systems, and obtains the result that directed union of systems is still a system. Based on the new concept of systems hereditary properties, discovers several hereditary properties of systems, and points out that the property of finite generativity of systems is not hereditary. Shows that the image and pre-image of subsystems under homomorphisms are still subsystems, and consequently several homomorphic invariable properties are obtained. Considers subsystems of induced systems and some applications of this research.