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

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July 20

Start anew: Embedding, monomorphism

I use the signs $\operatorname {dom} (t)$ and $\operatorname {im} (t),$ to indicate the domain and the image, of a given function $t,$ respectively.

By an "embedding" of a first given function $f:\operatorname {dom} (f)\to \operatorname {im} (f)$ in a second given function $g:\operatorname {dom} (g)\to \operatorname {im} (g),$

[or a "monomorphism" from a first given function $f:\operatorname {dom} (f)\to \operatorname {im} (f)$ to a second given function $g:\operatorname {dom} (g)\to \operatorname {im} (g)\ ],$

I mean, as expected:

a pair of one-to-one maps:

$\phi :\operatorname {dom} (f)\to \operatorname {dom} (g),$ i.e. from the domain of $f$ to the domain of $g,$

$\varPhi :\operatorname {im} (f)\to \operatorname {im} (g),$ i.e. from the image of $f$ to the image of $g,$

satisfying $:\ g\circ \phi =\varPhi \circ f.$

Question:

Given a pair of non-constant linear functions (i.e. functions of the form $px+q,$ with constants $p\neq 0,\ q,$ for each function):

$f:\operatorname {dom} (f)\to \operatorname {im} (f),$ having at least four elements in its domain $\operatorname {im} (f),$

$g:\operatorname {dom} (g)\to \operatorname {im} (g),$

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:

$F:\operatorname {im} (f)\to \operatorname {im} (F\circ f),$

$G:\operatorname {im} (g)\to \operatorname {im} (G\circ g),$

to satisfy the condition, that the first composition $(F\circ f):\operatorname {dom} (f)\to \operatorname {im} (F\circ f)$ can be embedded in the second composition $(G\circ g):\operatorname {dom} (g)\to \operatorname {im} (G\circ g)\ ?$

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 $F,G$ 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 $N:\operatorname {im} (F\circ f)\to \operatorname {im} (G\circ g),$ 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)

Could you make the domains and codomains of these functions explicit, like $f:A\to B$ ?  --Lambiam 21:39, 20 July 2023 (UTC)

Done. See above. 2A06:C701:7471:3000:39AA:1A85:25C2:975B (talk) 23:01, 20 July 2023 (UTC)