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4

$begingroup$

I need some help for one equality in the following proposition. It was a hint to conclude that $supf(cdot):finPhi$ is additive. I highlighted it blue. Ultimately I am interested in proving the proposition, but the main question is a hint with how to obtain the blue equality $overset☹=$.

Proposition. Let $(X,+,0)$ be an abelian monoid. Let $(M,preceq, +,0)$ be an ordered monoid. Fix $Phisubseteq M^X$. Assume the following.

(i) $(forall finPhi)[f:(X,+,0)to (M,+,0)]$

(ii) $(forall f_1,f_2inPhi)(exists f_3inPhi)(forall xin X)[f_1(x)preceq f_3(x)text and f_2(x)preceq f_3(x)]$

(iii) $(forall xin X)[supf(x):finPhitext exists in M]$

Then $supf(cdot):finPhi$ is additive.

hint provided from source material:
For all $x,yin X$,
$$beginalign
colorbluesupf(x):finPhi+supf(y):finPhi&oversetcolorblue☹=colorbluesupf_1(x)+f_2(y):f_1,f_2inPhi\
&preceq supf(x)+f(y):finPhi\
&=supf(x+y):finPhi\
&preceqsupf(x):finPhi+supf(y):finPhi
endalign$$

I can easily show $succeq$ of $overset☹=$, but not the other direction. I believe I am missing something subtle with the assumptions because with ordered monoids one has $$sup S+sup Tsucceq sup(S+T)tag1$$
when the suprema all exist. In particular, if one considers $(-infty,+infty]$ adjoined with $ncdot +infty_ninmathbbN$ we have an ordered monoid where if $S=T=mathbbR$, then we get a strict inequality of $(1)$. If instead one considers ordered groups, then one can show equality in $(1)$. It is because of this that I somehow believe the proposition should be considering ordered groups instead of ordered monoids, but, again, I may be missing something subtle with the assumptions.

I also want to add I already know how to prove the proposition if I have the blue equality at hand.

order-theory supremum-and-infimum monoid

share|cite|improve this question

edited 2 days ago

Alberto Takase

asked Mar 11 at 8:49

Alberto TakaseAlberto Takase

2,380719

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What blue equality? I didn't know that equalities could get the blues.
  $endgroup$
  – William Elliot
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @WilliamElliot I made some edits: $overset☹=$
  $endgroup$
  – Alberto Takase
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You have already given a counter example, so assume group.
  $endgroup$
  – William Elliot
  2 days ago
  

  
  
  
  

add a comment | 

4

$begingroup$

I need some help for one equality in the following proposition. It was a hint to conclude that $supf(cdot):finPhi$ is additive. I highlighted it blue. Ultimately I am interested in proving the proposition, but the main question is a hint with how to obtain the blue equality $overset☹=$.

Proposition. Let $(X,+,0)$ be an abelian monoid. Let $(M,preceq, +,0)$ be an ordered monoid. Fix $Phisubseteq M^X$. Assume the following.

(i) $(forall finPhi)[f:(X,+,0)to (M,+,0)]$

(ii) $(forall f_1,f_2inPhi)(exists f_3inPhi)(forall xin X)[f_1(x)preceq f_3(x)text and f_2(x)preceq f_3(x)]$

(iii) $(forall xin X)[supf(x):finPhitext exists in M]$

Then $supf(cdot):finPhi$ is additive.

hint provided from source material:
For all $x,yin X$,
$$beginalign
colorbluesupf(x):finPhi+supf(y):finPhi&oversetcolorblue☹=colorbluesupf_1(x)+f_2(y):f_1,f_2inPhi\
&preceq supf(x)+f(y):finPhi\
&=supf(x+y):finPhi\
&preceqsupf(x):finPhi+supf(y):finPhi
endalign$$

I can easily show $succeq$ of $overset☹=$, but not the other direction. I believe I am missing something subtle with the assumptions because with ordered monoids one has $$sup S+sup Tsucceq sup(S+T)tag1$$
when the suprema all exist. In particular, if one considers $(-infty,+infty]$ adjoined with $ncdot +infty_ninmathbbN$ we have an ordered monoid where if $S=T=mathbbR$, then we get a strict inequality of $(1)$. If instead one considers ordered groups, then one can show equality in $(1)$. It is because of this that I somehow believe the proposition should be considering ordered groups instead of ordered monoids, but, again, I may be missing something subtle with the assumptions.

I also want to add I already know how to prove the proposition if I have the blue equality at hand.

order-theory supremum-and-infimum monoid

share|cite|improve this question

edited 2 days ago

Alberto Takase

asked Mar 11 at 8:49

Alberto TakaseAlberto Takase

2,380719

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What blue equality? I didn't know that equalities could get the blues.
  $endgroup$
  – William Elliot
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @WilliamElliot I made some edits: $overset☹=$
  $endgroup$
  – Alberto Takase
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You have already given a counter example, so assume group.
  $endgroup$
  – William Elliot
  2 days ago
  

  
  
  
  

add a comment | 

$begingroup$

I need some help for one equality in the following proposition. It was a hint to conclude that $supf(cdot):finPhi$ is additive. I highlighted it blue. Ultimately I am interested in proving the proposition, but the main question is a hint with how to obtain the blue equality $overset☹=$.

Proposition. Let $(X,+,0)$ be an abelian monoid. Let $(M,preceq, +,0)$ be an ordered monoid. Fix $Phisubseteq M^X$. Assume the following.

(i) $(forall finPhi)[f:(X,+,0)to (M,+,0)]$

(ii) $(forall f_1,f_2inPhi)(exists f_3inPhi)(forall xin X)[f_1(x)preceq f_3(x)text and f_2(x)preceq f_3(x)]$

(iii) $(forall xin X)[supf(x):finPhitext exists in M]$

Then $supf(cdot):finPhi$ is additive.

hint provided from source material:
For all $x,yin X$,
$$beginalign
colorbluesupf(x):finPhi+supf(y):finPhi&oversetcolorblue☹=colorbluesupf_1(x)+f_2(y):f_1,f_2inPhi\
&preceq supf(x)+f(y):finPhi\
&=supf(x+y):finPhi\
&preceqsupf(x):finPhi+supf(y):finPhi
endalign$$

I can easily show $succeq$ of $overset☹=$, but not the other direction. I believe I am missing something subtle with the assumptions because with ordered monoids one has $$sup S+sup Tsucceq sup(S+T)tag1$$
when the suprema all exist. In particular, if one considers $(-infty,+infty]$ adjoined with $ncdot +infty_ninmathbbN$ we have an ordered monoid where if $S=T=mathbbR$, then we get a strict inequality of $(1)$. If instead one considers ordered groups, then one can show equality in $(1)$. It is because of this that I somehow believe the proposition should be considering ordered groups instead of ordered monoids, but, again, I may be missing something subtle with the assumptions.

I also want to add I already know how to prove the proposition if I have the blue equality at hand.

order-theory supremum-and-infimum monoid

share|cite|improve this question

edited 2 days ago

Alberto Takase

asked Mar 11 at 8:49

Alberto TakaseAlberto Takase

2,380719

$endgroup$

share|cite|improve this question

edited 2 days ago

Alberto Takase

asked Mar 11 at 8:49

Alberto TakaseAlberto Takase

2,380719

share|cite|improve this question

edited 2 days ago

Alberto Takase

asked Mar 11 at 8:49

Alberto TakaseAlberto Takase

2,380719

asked Mar 11 at 8:49

Alberto TakaseAlberto Takase

2,380719

asked Mar 11 at 8:49

Alberto TakaseAlberto Takase

2,380719