What do you call this homomorphism?

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In summary, the conversation discusses the concept of endomorphism, which is a type of homomorphism that maps elements in a set to themselves. The set of all homomorphisms is denoted as U, while the identity mapping is denoted as I. The complementary set of identity map is denoted as U-I. It is noted that in an endomorphism, the function f(a) is not equal to a for all elements in the set. The conversation also mentions that in a group homomorphism, the element 1 is always mapped to 1. The participants come to a conclusion that the given function is most likely an endomorphism.
  • #1
tgt
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2
f:A->A but f(a) does not equal a for all a in A.
 
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  • #2
Endomorphism?
 
  • #3
If set of all homomorphism is denoted as U,
and identity mapping is denoted as I

That is complementary set of identity map.
denoted as U-I
 
  • #4
So you have a "set of homomorphisms" minus a single homomorphism? U- {I} would make more sense. But I would interpret "does not equal a for all a in A" as meaning f(a) is NEVER equal to a.
 
  • #5
I am confused, too.
A group homomorphism always
maps 1 to 1, so...
 
  • #6
HallsofIvy said:
So you have a "set of homomorphisms" minus a single homomorphism? U- {I} would make more sense. But I would interpret "does not equal a for all a in A" as meaning f(a) is NEVER equal to a.

fair point. It's probably an endomorphism.
 
  • #7
waht said:
Endomorphism?

that seems right.
 

FAQ: What do you call this homomorphism?

What is a homomorphism?

A homomorphism is a mathematical function that preserves the algebraic structure of a set, meaning that the operation performed on the elements of the set remains unchanged after the function is applied.

What is the purpose of a homomorphism?

The purpose of a homomorphism is to study the relationship between two algebraic structures by mapping one structure to another in a way that preserves their essential properties.

What are some common examples of homomorphisms?

Some common examples of homomorphisms include the addition and multiplication functions on the real numbers, the logarithmic function, and the trigonometric functions.

What is the difference between a homomorphism and an isomorphism?

While homomorphisms preserve the algebraic structure of a set, isomorphisms preserve both the structure and the elements of a set. In other words, isomorphisms are bijective homomorphisms.

How are homomorphisms used in scientific research?

Homomorphisms are used in various fields of science, such as physics, chemistry, and biology, to study the relationships between different mathematical models and structures. They also play a crucial role in data analysis and machine learning algorithms.

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