Deductive systems of GE-algebras

A new sub-structure called (vivid) deductive system is introduced and their properties 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 deductive 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.