Doesn't the identity relation have to be a homogeneous relation?

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On the wikipedia page for converse relation it states:

If $I$ represents the identity relation, then a relation $R$ may have an inverse as follows:

A relation $R$ is called right-invertible if there exists a relation $X$ with $displaystyle Rcirc X=I$, and left-invertible if there exists a $Y$ with $displaystyle Ycirc R=I$. Then $X$ and $Y$ are called the right and left inverse of $R$, respectively. Right- and left-invertible relations are called invertible. For invertible homogeneous relations all right and left inverses coincide; the notion inverse $R^–1$ is used. Then $R^–1$ = $R^T$ holds.

My question is that doesn't the identity relation only make sense if it is a homogenous relation i.e: $Rsubset Stimes S$? If so then the above would suggest that all left and right inverses are equivalent, which doesn't seem right.

relations inverse

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edited Aug 3 at 11:28

Asaf Karagila

291k31401731

asked Aug 3 at 7:49

Ozaner Hansha

14912

add a comment | 

up vote
0
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On the wikipedia page for converse relation it states:

If $I$ represents the identity relation, then a relation $R$ may have an inverse as follows:

A relation $R$ is called right-invertible if there exists a relation $X$ with $displaystyle Rcirc X=I$, and left-invertible if there exists a $Y$ with $displaystyle Ycirc R=I$. Then $X$ and $Y$ are called the right and left inverse of $R$, respectively. Right- and left-invertible relations are called invertible. For invertible homogeneous relations all right and left inverses coincide; the notion inverse $R^–1$ is used. Then $R^–1$ = $R^T$ holds.

My question is that doesn't the identity relation only make sense if it is a homogenous relation i.e: $Rsubset Stimes S$? If so then the above would suggest that all left and right inverses are equivalent, which doesn't seem right.

relations inverse

share|cite|improve this question

edited Aug 3 at 11:28

Asaf Karagila

291k31401731

asked Aug 3 at 7:49

Ozaner Hansha

14912

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

On the wikipedia page for converse relation it states:

If $I$ represents the identity relation, then a relation $R$ may have an inverse as follows:

A relation $R$ is called right-invertible if there exists a relation $X$ with $displaystyle Rcirc X=I$, and left-invertible if there exists a $Y$ with $displaystyle Ycirc R=I$. Then $X$ and $Y$ are called the right and left inverse of $R$, respectively. Right- and left-invertible relations are called invertible. For invertible homogeneous relations all right and left inverses coincide; the notion inverse $R^–1$ is used. Then $R^–1$ = $R^T$ holds.

My question is that doesn't the identity relation only make sense if it is a homogenous relation i.e: $Rsubset Stimes S$? If so then the above would suggest that all left and right inverses are equivalent, which doesn't seem right.

relations inverse

share|cite|improve this question

edited Aug 3 at 11:28

Asaf Karagila

291k31401731

asked Aug 3 at 7:49

Ozaner Hansha

14912

On the wikipedia page for converse relation it states:

If $I$ represents the identity relation, then a relation $R$ may have an inverse as follows:

A relation $R$ is called right-invertible if there exists a relation $X$ with $displaystyle Rcirc X=I$, and left-invertible if there exists a $Y$ with $displaystyle Ycirc R=I$. Then $X$ and $Y$ are called the right and left inverse of $R$, respectively. Right- and left-invertible relations are called invertible. For invertible homogeneous relations all right and left inverses coincide; the notion inverse $R^–1$ is used. Then $R^–1$ = $R^T$ holds.

My question is that doesn't the identity relation only make sense if it is a homogenous relation i.e: $Rsubset Stimes S$? If so then the above would suggest that all left and right inverses are equivalent, which doesn't seem right.

relations inverse

share|cite|improve this question

edited Aug 3 at 11:28

Asaf Karagila

291k31401731

asked Aug 3 at 7:49

Ozaner Hansha

14912

share|cite|improve this question

share|cite|improve this question

edited Aug 3 at 11:28

Asaf Karagila

291k31401731

edited Aug 3 at 11:28

Asaf Karagila

291k31401731

edited Aug 3 at 11:28

Asaf Karagila

291k31401731

291k31401731

asked Aug 3 at 7:49

Ozaner Hansha

14912

asked Aug 3 at 7:49

Ozaner Hansha

14912

asked Aug 3 at 7:49

Ozaner Hansha

14912

14912

add a comment | 

add a comment | 

1 Answer
1

active

oldest

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up vote
1
down vote



accepted

Yes, that's true.

But there is always such set $S$. Simply take $S=operatornamedom(R)cupoperatornamerng(R)$, this is sometimes referred to as the field of $R$ or $operatornamefld(R)$.

If you have several relations, then you can simply take the union of their fields as your "set of interest".

share|cite|improve this answer

answered Aug 3 at 11:29

Asaf Karagila

291k31401731

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So then what's the point in left invertible relations vs. right invertible relations?
  – Ozaner Hansha
  Aug 3 at 12:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What do you mean?
  – Asaf Karagila
  Aug 3 at 12:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  "For invertible homogeneous relations, all right and left inverses coincide" and if we have to take the field of the relation to make sense of the identity relation, then the relation is homogeneous no? And if it is homogeneous then both the left and right inverses coincide if they exist.
  – Ozaner Hansha
  Aug 3 at 12:12
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Then they might talk about the identity relation on the domain and range of $R$. Flag this as unclear, I don't know what to tell you about the original intention of whoever wrote that paragraph on Wikipedia. But I can tell you that while in general Wikipedia tends to be a good source for mathematical definitions, it's usually much better to learn from books.
  – Asaf Karagila
  Aug 3 at 12:19
  

  
  
  
  

add a comment | 

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote



accepted

Yes, that's true.

But there is always such set $S$. Simply take $S=operatornamedom(R)cupoperatornamerng(R)$, this is sometimes referred to as the field of $R$ or $operatornamefld(R)$.

If you have several relations, then you can simply take the union of their fields as your "set of interest".

share|cite|improve this answer

answered Aug 3 at 11:29

Asaf Karagila

291k31401731

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So then what's the point in left invertible relations vs. right invertible relations?
  – Ozaner Hansha
  Aug 3 at 12:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What do you mean?
  – Asaf Karagila
  Aug 3 at 12:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  "For invertible homogeneous relations, all right and left inverses coincide" and if we have to take the field of the relation to make sense of the identity relation, then the relation is homogeneous no? And if it is homogeneous then both the left and right inverses coincide if they exist.
  – Ozaner Hansha
  Aug 3 at 12:12
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Then they might talk about the identity relation on the domain and range of $R$. Flag this as unclear, I don't know what to tell you about the original intention of whoever wrote that paragraph on Wikipedia. But I can tell you that while in general Wikipedia tends to be a good source for mathematical definitions, it's usually much better to learn from books.
  – Asaf Karagila
  Aug 3 at 12:19
  

  
  
  
  

add a comment | 

up vote
1
down vote



accepted

Yes, that's true.

But there is always such set $S$. Simply take $S=operatornamedom(R)cupoperatornamerng(R)$, this is sometimes referred to as the field of $R$ or $operatornamefld(R)$.

If you have several relations, then you can simply take the union of their fields as your "set of interest".

share|cite|improve this answer

answered Aug 3 at 11:29

Asaf Karagila

291k31401731

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So then what's the point in left invertible relations vs. right invertible relations?
  – Ozaner Hansha
  Aug 3 at 12:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What do you mean?
  – Asaf Karagila
  Aug 3 at 12:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  "For invertible homogeneous relations, all right and left inverses coincide" and if we have to take the field of the relation to make sense of the identity relation, then the relation is homogeneous no? And if it is homogeneous then both the left and right inverses coincide if they exist.
  – Ozaner Hansha
  Aug 3 at 12:12
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Then they might talk about the identity relation on the domain and range of $R$. Flag this as unclear, I don't know what to tell you about the original intention of whoever wrote that paragraph on Wikipedia. But I can tell you that while in general Wikipedia tends to be a good source for mathematical definitions, it's usually much better to learn from books.
  – Asaf Karagila
  Aug 3 at 12:19
  

  
  
  
  

add a comment | 

up vote
1
down vote



accepted

up vote
1
down vote



accepted

Yes, that's true.

But there is always such set $S$. Simply take $S=operatornamedom(R)cupoperatornamerng(R)$, this is sometimes referred to as the field of $R$ or $operatornamefld(R)$.

If you have several relations, then you can simply take the union of their fields as your "set of interest".

share|cite|improve this answer

answered Aug 3 at 11:29

Asaf Karagila

291k31401731

Yes, that's true.

But there is always such set $S$. Simply take $S=operatornamedom(R)cupoperatornamerng(R)$, this is sometimes referred to as the field of $R$ or $operatornamefld(R)$.

If you have several relations, then you can simply take the union of their fields as your "set of interest".

share|cite|improve this answer

answered Aug 3 at 11:29

Asaf Karagila

291k31401731

share|cite|improve this answer

share|cite|improve this answer

answered Aug 3 at 11:29

Asaf Karagila

291k31401731

answered Aug 3 at 11:29

Asaf Karagila

291k31401731

answered Aug 3 at 11:29

Asaf Karagila

291k31401731

291k31401731

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So then what's the point in left invertible relations vs. right invertible relations?
  – Ozaner Hansha
  Aug 3 at 12:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What do you mean?
  – Asaf Karagila
  Aug 3 at 12:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  "For invertible homogeneous relations, all right and left inverses coincide" and if we have to take the field of the relation to make sense of the identity relation, then the relation is homogeneous no? And if it is homogeneous then both the left and right inverses coincide if they exist.
  – Ozaner Hansha
  Aug 3 at 12:12
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Then they might talk about the identity relation on the domain and range of $R$. Flag this as unclear, I don't know what to tell you about the original intention of whoever wrote that paragraph on Wikipedia. But I can tell you that while in general Wikipedia tends to be a good source for mathematical definitions, it's usually much better to learn from books.
  – Asaf Karagila
  Aug 3 at 12:19
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So then what's the point in left invertible relations vs. right invertible relations?
  – Ozaner Hansha
  Aug 3 at 12:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What do you mean?
  – Asaf Karagila
  Aug 3 at 12:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  "For invertible homogeneous relations, all right and left inverses coincide" and if we have to take the field of the relation to make sense of the identity relation, then the relation is homogeneous no? And if it is homogeneous then both the left and right inverses coincide if they exist.
  – Ozaner Hansha
  Aug 3 at 12:12
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Then they might talk about the identity relation on the domain and range of $R$. Flag this as unclear, I don't know what to tell you about the original intention of whoever wrote that paragraph on Wikipedia. But I can tell you that while in general Wikipedia tends to be a good source for mathematical definitions, it's usually much better to learn from books.
  – Asaf Karagila
  Aug 3 at 12:19
  

  
  
  
  

So then what's the point in left invertible relations vs. right invertible relations?
– Ozaner Hansha
Aug 3 at 12:06

So then what's the point in left invertible relations vs. right invertible relations?
– Ozaner Hansha
Aug 3 at 12:06

What do you mean?
– Asaf Karagila
Aug 3 at 12:06

What do you mean?
– Asaf Karagila
Aug 3 at 12:06

"For invertible homogeneous relations, all right and left inverses coincide" and if we have to take the field of the relation to make sense of the identity relation, then the relation is homogeneous no? And if it is homogeneous then both the left and right inverses coincide if they exist.
– Ozaner Hansha
Aug 3 at 12:12

"For invertible homogeneous relations, all right and left inverses coincide" and if we have to take the field of the relation to make sense of the identity relation, then the relation is homogeneous no? And if it is homogeneous then both the left and right inverses coincide if they exist.
– Ozaner Hansha
Aug 3 at 12:12

1

1

Then they might talk about the identity relation on the domain and range of $R$. Flag this as unclear, I don't know what to tell you about the original intention of whoever wrote that paragraph on Wikipedia. But I can tell you that while in general Wikipedia tends to be a good source for mathematical definitions, it's usually much better to learn from books.
– Asaf Karagila
Aug 3 at 12:19

Then they might talk about the identity relation on the domain and range of $R$. Flag this as unclear, I don't know what to tell you about the original intention of whoever wrote that paragraph on Wikipedia. But I can tell you that while in general Wikipedia tends to be a good source for mathematical definitions, it's usually much better to learn from books.
– Asaf Karagila
Aug 3 at 12:19

add a comment | 

 

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