deductive systems of ge-algebras

A new sub-structure called (vivid) deductive system is introduced and their prop- erties are examined. conditions for a subset to be a deductive system are provided. the notion of upper ge-set is also introduced, and an example to show that any upper ge-set may not be a deductive system are supplied. conditions for an upper ge-set to be a deductive system are provided. an upper ge-set is used to consider conditions for a subset to be a deductive system. the characterization of deductive system is established, and relationship between de- ductive system and vivid deductive system are created. conditions for a deductive system to be a vivid deductive system are given, and the extension property for vivid deductive system is constructed.