Terminology involving "homomorphic"

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In summary, several current Wikipedia articles on "homomorphism" avoid using the adjective "homomorphic" due to the potential ambiguity of its usage. It is suggested to use "A is homomorphic to B" to mean "There is a homomorphism from B to A", although this relation is not symmetric. The term "homomorphism" is also used to mean "A is homomorphic to B". However, this definition may lead to confusion and should be avoided. The terminology of "there is a homomorphism of A _into_ B" is proposed as a clearer alternative. In addition, a homomorphism from A onto B is called an epimorphism, and an embedding or monomorphism is used to describe
  • #1
Stephen Tashi
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I notice several current Wikipedia articles on "homomorphism" avoid using the adjective "homomorphic". (e.g. https://en.wikipedia.org/wiki/Group_homomorphism , https://en.wikipedia.org/wiki/Homomorphism).

Of course the problem with saying "A and B are homomorphic" is that there can be a homomorphism from A to B but no homomorphism from B to A may exist. So it would seem best to only to use the terminology "A is homomorphic to B", which suggests the possibility of an asymmetrical relation. But what do we want that terminology to mean? The natural definition (in my opinion) is that it should mean there is a homomorphism from B to A. However, I've heard people use it to mean that there is a homomorphism from A to B.
 
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  • #2
Why would you want to use the word homomorphic?!
 
  • #3
martinbn said:
Why would you want to use the word homomorphic?!

For example, if ##A## is a ##p##-group, and ##A## is homomorphic to ##B##, then there is a subgroup of ##B## such that ##B## is a ##p##-group (this subgroup is the image of the homomorphism). I do agree that the terminology is ambiguous, and should therefore be avoided.
 
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  • #4
So if "A is homomorphic to B" means "There is a homomorpism from A to B", then this relation is not symmetric. For example (as fields), ##\mathbb Q## is homomorphic to ##\mathbb R## (take the inclusion map as the homomorphism), but not the other way round. This is a difference from the case with "isomorphic" and "isomorphism".
 
  • #5
Erland said:
For example (as fields), ##\mathbb Q## is homomorphic to ##\mathbb R## (take the inclusion map as the homomorphism), but not the other way round.

Couldn't we map the all the reals to the identity element? The identity element is a subset of ##\mathbb Q##, so that would be a homomorphism from ##\mathbb R## to ##\mathbb Q##.
 
  • #6
Isn't any group homomorphic to any other group if we pick the trivial map?
That would make the concept meaningless.
 
  • #7
Maybe we should use the expression "there is a homomorphism of A _into_ B " , as opposed to a homomorphism of A _onto_ B.
Similarly, there may be homeomorphisms _into_ ( meaning embeddings) without homeomorphisms _onto_. I EDIT :AFAIK, a bijective homomorphism is an isomorphism , though the equivalent is not true for hemeomorphisms.
 
  • #8
Stephen Tashi said:
Couldn't we map the all the reals to the identity element? The identity element is a subset of ##\mathbb Q##, so that would be a homomorphism from ##\mathbb R## to ##\mathbb Q##.
Yes, you are right!
 
  • #9
I like Serena said:
Isn't any group homomorphic to any other group if we pick the trivial map?
That would make the concept meaningless.
Exactly!
 
  • #10
WWGD said:
Maybe we should use the expression "there is a homomorphism of A _into_ B " , as opposed to a homomorphism of A _onto_ B.
Similarly, there may be homeomorphisms _into_ ( meaning embeddings) without homeomorphisms _onto_. I EDIT :AFAIK, a bijective homomorphism is an isomorphism , though the equivalent is not true for hemeomorphisms.
A homomorphism from A onto B is called an epimorphism.
 
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  • #11
Erland said:
A homomorphism from A onto B is called an epimorphism.
Still, unfortunately many are not precise in this usage. Isn't this called an embedding in this " category" (Algebraic objects: rings, groups, etc.) so that A embeds in B?
 
  • #12
  • #13
Erland said:
Not in general. To call a map an embedding it must be one-to-one (injective). In algebra, an embedding is also called a monomorphism.
https://en.wikipedia.org/wiki/Embedding
Yes, sorry, this is what I was assuming, and it is a stronger condition, meaning there is a "copy" of A living in B , much more stringent than just having a homomorphism..
 
  • #14
On second thought, it there is technicality in the definition of a "field homomorphism" that would disqualify mapping all of ##Q## to the multiplicative identity as a homomorphism between fields. http://planetmath.org/fieldhomomorphism
 
  • #15
Stephen Tashi said:
On second thought, it there is technicality in the definition of a "field homomorphism" that would disqualify mapping all of ##Q## to the multiplicative identity as a homomorphism between fields. http://planetmath.org/fieldhomomorphism
Ah yes, so a field homomorphism from ##\mathbb Q \to \mathbb R## has to be the canonical map, doesn't it?
And more generally, a field homomorphism has to be from the 'smaller' field to the 'larger' field, effectively eliminating the ambiguity of direction.
 
  • #16
Just curious, in the finite case, one obstruction to the existence of non-trivial ( non-identity) homomorphisms f
between finite rings A,B is that f(A) must be a subgroup/ring/etc. and then there are divisibility issues (right)?
Still, what are the obstructions for the existence of homomorphisms between infinite rings? I remember a similar
question in topology about the existence of continuous maps, and the answer was that of topological invariants.
(e.g., there cannot be a continuous , non-constant map between a connected space and a disconnected space).
What are the obstructions between infinite algebraic objects?
 
  • #17
WWGD said:
Just curious, in the finite case, one obstruction to the existence of non-trivial ( non-identity) homomorphisms f
between finite rings A,B is that f(A) must be a subgroup/ring/etc. and then there are divisibility issues (right)?
Still, what are the obstructions for the existence of homomorphisms between infinite rings? I remember a similar
question in topology about the existence of continuous maps, and the answer was that of topological invariants.
(e.g., there cannot be a continuous , non-constant map between a connected space and a disconnected space).
What are the obstructions between infinite algebraic objects?
Just like a field homomorphism, a ring homomorphism cannot be trivial - unless we have the zero ring with 1 element (0=1) as codomain.
 
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  • #18
all i can think of say for commutative rings with identity, and which take identity to identity, is that some quotient of the source must embed in the target. in particular for fields, since they have no non trivial quotients, i.e. no proper ideals, the source itself must embed in the target. the term is not much needed in general since we already have terms for the existence of surjective and injective homomorphisms, i.e. (isomorphism with) a quotient or subobject.
 
  • #19
I like Serena said:
Just like a field homomorphism, a ring homomorphism cannot be trivial - unless we have the zero ring with 1 element (0=1) as codomain.
Still, as an example of a non-trivial one between a ring without unity and one with unity, we have the inclusion of ## 2\mathbb Z \rightarrow \mathbb Z ##.
 
  • #20
I like Serena said:
Just like a field homomorphism, a ring homomorphism cannot be trivial - unless we have the zero ring with 1 element (0=1) as codomain.
My bad, I am used to dealing with simpler objects like vector spaces , with a single structure ( addition of vectors) where we may map into ( additive) 0.
 

Related to Terminology involving "homomorphic"

1. What is the definition of homomorphic terminology?

Homomorphic terminology refers to words or phrases used in the field of mathematics and computer science that describe the properties and operations of homomorphic encryption. Homomorphic encryption is a type of encryption that allows for computations to be performed on encrypted data without the need for decryption.

2. How does homomorphic terminology differ from traditional encryption terminology?

Homomorphic terminology is specific to the field of homomorphic encryption and is used to describe concepts and operations unique to this type of encryption. Traditional encryption terminology may differ in that it focuses on other types of encryption, such as symmetric or asymmetric encryption.

3. What are some examples of homomorphic terminology?

Some examples of homomorphic terminology include "ciphertext," which refers to encrypted data, and "plaintext," which refers to unencrypted data. Other examples include "encryption scheme," "key generation," and "decryption function."

4. How is homomorphic terminology used in real-world applications?

Homomorphic terminology is used in real-world applications of homomorphic encryption, such as secure cloud computing and data privacy. It allows for sensitive data to be encrypted and processed in the cloud without the need for decryption, providing an extra layer of security and privacy for users.

5. Are there any challenges associated with understanding homomorphic terminology?

As with any specialized field, understanding homomorphic terminology may present challenges for those who are not familiar with the concepts and principles of homomorphic encryption. However, with proper education and training, these challenges can be overcome to fully understand and utilize homomorphic terminology in research and applications.

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