A definition of %structure kind name% homomorphism
Topics
About: structure
The table of contents of this article
Starting Context
- The reader knows a definition of structure.
- The reader knows a definition of map.
Target Context
- The reader will have a definition of %structure kind name% homomorphism.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Definition
For any structures of the same kind, \(S_1\) and \(S_2\), any map, \(f: S_1 \to S_2\), by that the image of any construction of any elements is the same construction of the images of the elements
"construction" there means a construction specific for the kind of the structures, for example, \(g_1 g_2\) for a group, but \(a_1 v_1 + a_2 v_2\) for a vectors space.
The definition means, for example, \(f (g_1 g_2) = f (g_1) f (g_2)\) for group homomorphism and \(f (a_1 v_1 + a_2 v_2) = a_1 f (v_1) + a_2 f (v_2)\) for vectors space homomorphism.
2: Note
Often called just "homomorphism", any map is never just a "homomorphism", but a group homomorphism, a vectors space homomorphism, etc., depending on what kind of structures, the domain and the codomain of the map are regarded to be of, which is the reason why the title of this article is "%structure kind name% homomorphism" with "%structure kind name%" as a place holder.
For example, when the domain, \(S_1\), and the codomain, \(S_2\), are vectors spaces (which means that they are automatically also groups with addition as the group operation), a map may be a group homomorphism, but not a vectors space homomorphism.