Wikipedia:Reference desk/Archives/Mathematics/2023 July 20

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July 20

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I use the signs   and   to indicate the domain and the image, of a given function   respectively.

By an "embedding" of a first given function   in a second given function  

[or a "monomorphism" from a first given function   to a second given function  

I mean, as expected:

a pair of one-to-one maps:

  i.e. from the domain of   to the domain of  
  i.e. from the image of   to the image of  

satisfying  

Question:

Given a pair of non-constant linear functions (i.e. functions of the form   with constants   for each function):

  having at least four elements in its domain  
 

of which the first function can be embedded in the second function,

what condition, being more intuitive than the following one, and being as less restrictive as possible (i.e. as weak as possible), is sufficient for a given additional pair of functions:

 
 

to satisfy the condition, that the first composition   can be embedded in the second composition  

Terminological clarification: By an "as less restrictive condition as possible (i.e. as weak as possible)", the purpose is, for example, to prevent the condition from requiring that two given additional functions   be the identity function, and in particular to prevent the condition from posing in advance any restriction on the size of a the image of a given composition.

P.S. Of course, such a more intuitive sufficient condition will have to assume (at least implicitly) some necessary preconditions, e.g. that there is a one-to-one map   i.e. from the image of the first composition to the image of the second composition.

2A06:C701:7471:3000:39AA:1A85:25C2:975B (talk) 17:01, 20 July 2023 (UTC)[reply]

Could you make the domains and codomains of these functions explicit, like   ?  --Lambiam 21:39, 20 July 2023 (UTC)[reply]
Done. See above. 2A06:C701:7471:3000:39AA:1A85:25C2:975B (talk) 23:01, 20 July 2023 (UTC)[reply]