What do you call this homomorphism?

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Discussion Overview

The discussion revolves around the classification of a specific type of homomorphism, particularly focusing on a function f: A -> A that does not map any element a in A to itself. Participants explore the implications of this definition and seek to identify the correct terminology for such a mapping.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant suggests that the function f could be called an endomorphism.
  • Another participant proposes that the set of all homomorphisms, denoted as U, minus the identity mapping, denoted as I, could represent this function.
  • A different viewpoint questions whether the notation U - I is appropriate, suggesting that U - {I} would make more sense, while interpreting the condition "does not equal a for all a in A" as meaning f(a) is never equal to a.
  • One participant expresses confusion regarding the properties of group homomorphisms, noting that they always map the identity element to itself.
  • There is a reiteration of the endomorphism classification, with some participants agreeing that it seems to fit the definition.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate terminology and notation for the homomorphism in question. There is no consensus on a single definition or classification, as some participants support the endomorphism label while others raise questions about the implications of the mapping.

Contextual Notes

There are unresolved assumptions regarding the properties of the homomorphism and the implications of the definitions used. The discussion reflects varying interpretations of the conditions set forth in the initial post.

tgt
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f:A->A but f(a) does not equal a for all a in A.
 
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Endomorphism?
 
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
 
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.
 
I am confused, too.
A group homomorphism always
maps 1 to 1, so...
 
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.
 
what said:
Endomorphism?

that seems right.
 

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