Inheritance theory of ontologies is a very important unsolved problem.By extending the inheritance mathematical theory

the semantics of ontology inheritance and the mechanism of ontology inheritance are analyzed.The creative works are as followed:(1)The mathematics of inheritance systems isn't powerful enough to explain the ontology inheritance system.The "False conflict" will occur when we use the old theory of inheritance to describe the inheritance of class/property with exceptions.In order to solve this problem

the old theory is extended by adding tokens and redefining the definitions of conclusion set

inheritable and grounded expansion

etc.(2)Most theorems and corollaries prove to be correct on the new definitions.Above these new definitions

some new theorems and corollaries are got.(3)Transferring the representation of ontology (D

P

U

V

H) into triple (F

E

I).And proving the capability of the triple representation equals to the quintuple representation.The ontology is mapped into the extended inheritance system.The problem of consistency and un-ambiguity of ontology inheritance are explored with the extended theory.(4)The up-scan and down-scan algorithms are designed to detect the conflict in order to maintain the consistency of the system.The analysis of consistency of the ontology inheritance system effectively supports the ontology-oriented engineering methodologies booming now.