The name “section” for the operation of selecting representatives of an equivalence class
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This is a question about terminology and sources. While looking for a name for the operation of "picking a representative from an equivalence class", I came upon the Wikipedia article on equivalence classes, where this operation is referred to as a "section".
With this term, given an equivalence relation $sim$ on a set $X$, I can define a "canonical section" $s_c$ as an injective map $s_c colon X/sim to X$ and then add that the image by $s_c$ of an equivalence class is called its "canonical representative". This notation comes in very handy for formal derivations, where I can use $s_c(C)$ rather than natural-language descriptions like "and then we take the canonical representative of C,...".
But this use of the term "section" doesn't seem to be common at all in the textbooks about algebra, so I am not sure if I should use it in my current research. I think it comes from the use of "section" and "retraction" in category theory, but would it be understood by people with a background in other fields of mathematics? Is there any well-known textbook dealing with equivalence relations where the word "section" is used in this way?
abstract-algebra terminology equivalence-relations quotient-spaces
asked Dec 31 '18 at 11:18
Ángel José RiesgoÁngel José Riesgo
303
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add a comment |
$begingroup$
This is a question about terminology and sources. While looking for a name for the operation of "picking a representative from an equivalence class", I came upon the Wikipedia article on equivalence classes, where this operation is referred to as a "section".
With this term, given an equivalence relation $sim$ on a set $X$, I can define a "canonical section" $s_c$ as an injective map $s_c colon X/sim to X$ and then add that the image by $s_c$ of an equivalence class is called its "canonical representative". This notation comes in very handy for formal derivations, where I can use $s_c(C)$ rather than natural-language descriptions like "and then we take the canonical representative of C,...".
But this use of the term "section" doesn't seem to be common at all in the textbooks about algebra, so I am not sure if I should use it in my current research. I think it comes from the use of "section" and "retraction" in category theory, but would it be understood by people with a background in other fields of mathematics? Is there any well-known textbook dealing with equivalence relations where the word "section" is used in this way?
abstract-algebra terminology equivalence-relations quotient-spaces
asked Dec 31 '18 at 11:18
Ángel José RiesgoÁngel José Riesgo
303
$endgroup$
3
$begingroup$
The term 'section' is quite common. The picture you want to have in your mind is the following: for a surjective function $f:Eto B$, each of the sets $f^{-1}(b)$ is nonempty, and draws a tower pancakes (or crepes if you're more into the savory stuff?) at a point $b$. Picking a section of $f$ amounts to cutting these towers all at once horizontally, effectively picking a cross-section of the tower.
$endgroup$
– Pedro Tamaroff♦
Dec 31 '18 at 16:23
add a comment |
$begingroup$
This is a question about terminology and sources. While looking for a name for the operation of "picking a representative from an equivalence class", I came upon the Wikipedia article on equivalence classes, where this operation is referred to as a "section".
With this term, given an equivalence relation $sim$ on a set $X$, I can define a "canonical section" $s_c$ as an injective map $s_c colon X/sim to X$ and then add that the image by $s_c$ of an equivalence class is called its "canonical representative". This notation comes in very handy for formal derivations, where I can use $s_c(C)$ rather than natural-language descriptions like "and then we take the canonical representative of C,...".
But this use of the term "section" doesn't seem to be common at all in the textbooks about algebra, so I am not sure if I should use it in my current research. I think it comes from the use of "section" and "retraction" in category theory, but would it be understood by people with a background in other fields of mathematics? Is there any well-known textbook dealing with equivalence relations where the word "section" is used in this way?
abstract-algebra terminology equivalence-relations quotient-spaces
asked Dec 31 '18 at 11:18
Ángel José RiesgoÁngel José Riesgo
303
$endgroup$
This is a question about terminology and sources. While looking for a name for the operation of "picking a representative from an equivalence class", I came upon the Wikipedia article on equivalence classes, where this operation is referred to as a "section".
With this term, given an equivalence relation $sim$ on a set $X$, I can define a "canonical section" $s_c$ as an injective map $s_c colon X/sim to X$ and then add that the image by $s_c$ of an equivalence class is called its "canonical representative". This notation comes in very handy for formal derivations, where I can use $s_c(C)$ rather than natural-language descriptions like "and then we take the canonical representative of C,...".
But this use of the term "section" doesn't seem to be common at all in the textbooks about algebra, so I am not sure if I should use it in my current research. I think it comes from the use of "section" and "retraction" in category theory, but would it be understood by people with a background in other fields of mathematics? Is there any well-known textbook dealing with equivalence relations where the word "section" is used in this way?
abstract-algebra terminology equivalence-relations quotient-spaces
abstract-algebra terminology equivalence-relations quotient-spaces
asked Dec 31 '18 at 11:18
Ángel José RiesgoÁngel José Riesgo
303
asked Dec 31 '18 at 11:18
Ángel José RiesgoÁngel José Riesgo
303
asked Dec 31 '18 at 11:18
Ángel José RiesgoÁngel José Riesgo
303
asked Dec 31 '18 at 11:18
Ángel José RiesgoÁngel José Riesgo
303
asked Dec 31 '18 at 11:18
Ángel José RiesgoÁngel José Riesgo
303
303
3
$begingroup$
The term 'section' is quite common. The picture you want to have in your mind is the following: for a surjective function $f:Eto B$, each of the sets $f^{-1}(b)$ is nonempty, and draws a tower pancakes (or crepes if you're more into the savory stuff?) at a point $b$. Picking a section of $f$ amounts to cutting these towers all at once horizontally, effectively picking a cross-section of the tower.
$endgroup$
– Pedro Tamaroff♦
Dec 31 '18 at 16:23
add a comment |
3
$begingroup$
The term 'section' is quite common. The picture you want to have in your mind is the following: for a surjective function $f:Eto B$, each of the sets $f^{-1}(b)$ is nonempty, and draws a tower pancakes (or crepes if you're more into the savory stuff?) at a point $b$. Picking a section of $f$ amounts to cutting these towers all at once horizontally, effectively picking a cross-section of the tower.
$endgroup$
– Pedro Tamaroff♦
Dec 31 '18 at 16:23
3
3
$begingroup$
The term 'section' is quite common. The picture you want to have in your mind is the following: for a surjective function $f:Eto B$, each of the sets $f^{-1}(b)$ is nonempty, and draws a tower pancakes (or crepes if you're more into the savory stuff?) at a point $b$. Picking a section of $f$ amounts to cutting these towers all at once horizontally, effectively picking a cross-section of the tower.
$endgroup$
– Pedro Tamaroff♦
Dec 31 '18 at 16:23
$begingroup$
The term 'section' is quite common. The picture you want to have in your mind is the following: for a surjective function $f:Eto B$, each of the sets $f^{-1}(b)$ is nonempty, and draws a tower pancakes (or crepes if you're more into the savory stuff?) at a point $b$. Picking a section of $f$ amounts to cutting these towers all at once horizontally, effectively picking a cross-section of the tower.
$endgroup$
– Pedro Tamaroff♦
Dec 31 '18 at 16:23
add a comment |
0
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$begingroup$
The term 'section' is quite common. The picture you want to have in your mind is the following: for a surjective function $f:Eto B$, each of the sets $f^{-1}(b)$ is nonempty, and draws a tower pancakes (or crepes if you're more into the savory stuff?) at a point $b$. Picking a section of $f$ amounts to cutting these towers all at once horizontally, effectively picking a cross-section of the tower.
$endgroup$
– Pedro Tamaroff♦
Dec 31 '18 at 16:23