definition of %structure kind name% homomorphism
Topics
About: structure
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
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: Structured Description
Here is the rules of Structured Description.
Entities:
\( S\): \(\in \{\text{ the structures kinds }\}\)
\( S_1\): \(\in \{\text{ the structures of the } S \text{ kind }\}\)
\( S_2\): \(\in \{\text{ the structures of the } S \text{ kind }\}\)
\(*f\): \(S_1 \to S_2\)
//
Conditions:
\(f\) preserves the structure of \(S_1\) in \(S_2\).
//
The details of "\(f\) preserves the structure of \(S_1\) in \(S_2\)" depend on the structure, but they are what can be successfully guessed (at least mostly); for example, when \(S\) is group, it means that the product of any elements is mapped to the product of the images of the elements, the identity element is mapped to the identity element, and the inverse of any element is mapped to the inverse of the image of the element.
2: Natural Language Description
For any structures kind, \(S\), any structure of the \(S\) kind, \(S_1\), and any structure of the \(S\) kind, \(S_2\), any map, \(f: S_1 \to S_2\), that preserves the structure of \(S_1\) in \(S_2\)
3: Note
A frequently seen definition of group homomorphism requires only that the product of any elements is mapped to the product of the images of the elements, which is because that guarantees that the identity element is mapped to the identity element and that the inverse of any element is mapped to the inverse of the image of the element because of the natures of group, but we prefer including all the requirements in the definition. That is because the general concept of 'homomorphism' requires all the requirements, and if group happens to have the natures that only a part of the requirements guarantees the rests, that is just a result. We can have the definition of group homomorphism with the full requirements and the proposition that a part of the requirements guarantees the rests, which seems a more appropriate argument.
As another example, a frequently seen definition of ring homomorphism does not require that the additive identity element is mapped to the additive identity element but requires that the multiplicative identity element is mapped to the multiplicative identity element, which is because the former requirement is automatically guaranteed while the latter requirement is not, but it should not be really a matter of that the definition does not require the former requirement, but is a matter of that there is a proposition that the former requirement is automatically guaranteed.
Of course, the frequently seen definitions do make any practical problem, but the way of thinking seems important.
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.
