first isomorphism theorem
Let be a fixed signature, and and structures for . If is a homomorphism, then there is a unique bimorphism such that for all , . Furthermore, if has the additional property that for each and each -ary relation symbol of ,
then is an isomorphism.
Since the homomorphic image of a -structure is also a -structure, we may assume that .
Let . Define a bimorphism . To verify that is well defined, let . Then . To show that is injective, suppose . Then , so . Hence . To show that is a homomorphism, observe that for any constant symbol of we have . For each and each -ary function symbol of ,
For each and each -ary relation symbol of ,
Thus is a bimorphism.
Now suppose has the additional property mentioned in the statement of the theorem. Then
Thus is an isomorphism. ∎
|Title||first isomorphism theorem|
|Date of creation||2013-03-22 13:50:42|
|Last modified on||2013-03-22 13:50:42|
|Last modified by||almann (2526)|