Question about the definition of left/right cosets
Let $(A,cdot)$ be a Magma. One defines another magma $(mathcal{P}(A),cdot)$ by letting
$forall_{B,Cinmathcal{P}(A)}Bcdot C :={bcdot c :bin B$ and $cin C}subseteq A.$
The case where B or C is a singleton set is of particulair interest and one then unses the following simplified Notation:
$forall_{Bin mathcal{P}(A)}forall_{ain A}(a cdot B := {a}cdot B, B cdot a := B cdot{a}).$
Sets of the form $acdot B$ are called $left$ $cosets$, sets of the form $B cdot a$ are called $right$ $cosets$
My Question is if you take for example the natural Numbers $mathbb{N}$ with a Operation defined as $cdot$ sucht that $(mathbb{N},cdot)$ is a magma and you show that ${1}cdot Bsubseteq mathbb{N}$, for some subset of $mathbb{N}$, is $1 cdot mathbb{N}$ then a left coset of $mathbb{N}$ or do you have to show that ${a}cdot Bsubseteq mathbb{N}$ holds true for any $ain mathbb{N}$?
abstract-algebra
asked Nov 27 '18 at 18:37
RM777
1978
add a comment |
Let $(A,cdot)$ be a Magma. One defines another magma $(mathcal{P}(A),cdot)$ by letting
$forall_{B,Cinmathcal{P}(A)}Bcdot C :={bcdot c :bin B$ and $cin C}subseteq A.$
The case where B or C is a singleton set is of particulair interest and one then unses the following simplified Notation:
$forall_{Bin mathcal{P}(A)}forall_{ain A}(a cdot B := {a}cdot B, B cdot a := B cdot{a}).$
Sets of the form $acdot B$ are called $left$ $cosets$, sets of the form $B cdot a$ are called $right$ $cosets$
My Question is if you take for example the natural Numbers $mathbb{N}$ with a Operation defined as $cdot$ sucht that $(mathbb{N},cdot)$ is a magma and you show that ${1}cdot Bsubseteq mathbb{N}$, for some subset of $mathbb{N}$, is $1 cdot mathbb{N}$ then a left coset of $mathbb{N}$ or do you have to show that ${a}cdot Bsubseteq mathbb{N}$ holds true for any $ain mathbb{N}$?
abstract-algebra
asked Nov 27 '18 at 18:37
RM777
1978
It's not very clear what you are asking: if $cdot$ is an operation on $Bbb{N}$, then $a cdot B subseteq Bbb{N}$ for any $a$ autormatically and $1 cdot Bbb{N}$ is a coset by definition. So what are you trying to prove?
– Rob Arthan
Nov 27 '18 at 18:47
For example let $cdot$ be the usual multiplication and $B={2,3}$, $1cdot B$ would be equal to the set ${2,3}$ is this set then by definition a left coset? Or is the set $acdot B,ainmathbb{N} = {2,3,4,6,8,9,12,....}$ a left coset?
– RM777
Nov 27 '18 at 19:01
add a comment |
Let $(A,cdot)$ be a Magma. One defines another magma $(mathcal{P}(A),cdot)$ by letting
$forall_{B,Cinmathcal{P}(A)}Bcdot C :={bcdot c :bin B$ and $cin C}subseteq A.$
The case where B or C is a singleton set is of particulair interest and one then unses the following simplified Notation:
$forall_{Bin mathcal{P}(A)}forall_{ain A}(a cdot B := {a}cdot B, B cdot a := B cdot{a}).$
Sets of the form $acdot B$ are called $left$ $cosets$, sets of the form $B cdot a$ are called $right$ $cosets$
My Question is if you take for example the natural Numbers $mathbb{N}$ with a Operation defined as $cdot$ sucht that $(mathbb{N},cdot)$ is a magma and you show that ${1}cdot Bsubseteq mathbb{N}$, for some subset of $mathbb{N}$, is $1 cdot mathbb{N}$ then a left coset of $mathbb{N}$ or do you have to show that ${a}cdot Bsubseteq mathbb{N}$ holds true for any $ain mathbb{N}$?
abstract-algebra
asked Nov 27 '18 at 18:37
RM777
1978
Let $(A,cdot)$ be a Magma. One defines another magma $(mathcal{P}(A),cdot)$ by letting
$forall_{B,Cinmathcal{P}(A)}Bcdot C :={bcdot c :bin B$ and $cin C}subseteq A.$
The case where B or C is a singleton set is of particulair interest and one then unses the following simplified Notation:
$forall_{Bin mathcal{P}(A)}forall_{ain A}(a cdot B := {a}cdot B, B cdot a := B cdot{a}).$
Sets of the form $acdot B$ are called $left$ $cosets$, sets of the form $B cdot a$ are called $right$ $cosets$
My Question is if you take for example the natural Numbers $mathbb{N}$ with a Operation defined as $cdot$ sucht that $(mathbb{N},cdot)$ is a magma and you show that ${1}cdot Bsubseteq mathbb{N}$, for some subset of $mathbb{N}$, is $1 cdot mathbb{N}$ then a left coset of $mathbb{N}$ or do you have to show that ${a}cdot Bsubseteq mathbb{N}$ holds true for any $ain mathbb{N}$?
abstract-algebra
abstract-algebra
asked Nov 27 '18 at 18:37
RM777
1978
asked Nov 27 '18 at 18:37
RM777
1978
asked Nov 27 '18 at 18:37
RM777
1978
asked Nov 27 '18 at 18:37
RM777
1978
asked Nov 27 '18 at 18:37
RM777
1978
1978
It's not very clear what you are asking: if $cdot$ is an operation on $Bbb{N}$, then $a cdot B subseteq Bbb{N}$ for any $a$ autormatically and $1 cdot Bbb{N}$ is a coset by definition. So what are you trying to prove?
– Rob Arthan
Nov 27 '18 at 18:47
For example let $cdot$ be the usual multiplication and $B={2,3}$, $1cdot B$ would be equal to the set ${2,3}$ is this set then by definition a left coset? Or is the set $acdot B,ainmathbb{N} = {2,3,4,6,8,9,12,....}$ a left coset?
– RM777
Nov 27 '18 at 19:01
add a comment |
It's not very clear what you are asking: if $cdot$ is an operation on $Bbb{N}$, then $a cdot B subseteq Bbb{N}$ for any $a$ autormatically and $1 cdot Bbb{N}$ is a coset by definition. So what are you trying to prove?
– Rob Arthan
Nov 27 '18 at 18:47
For example let $cdot$ be the usual multiplication and $B={2,3}$, $1cdot B$ would be equal to the set ${2,3}$ is this set then by definition a left coset? Or is the set $acdot B,ainmathbb{N} = {2,3,4,6,8,9,12,....}$ a left coset?
– RM777
Nov 27 '18 at 19:01
It's not very clear what you are asking: if $cdot$ is an operation on $Bbb{N}$, then $a cdot B subseteq Bbb{N}$ for any $a$ autormatically and $1 cdot Bbb{N}$ is a coset by definition. So what are you trying to prove?
– Rob Arthan
Nov 27 '18 at 18:47
It's not very clear what you are asking: if $cdot$ is an operation on $Bbb{N}$, then $a cdot B subseteq Bbb{N}$ for any $a$ autormatically and $1 cdot Bbb{N}$ is a coset by definition. So what are you trying to prove?
– Rob Arthan
Nov 27 '18 at 18:47
For example let $cdot$ be the usual multiplication and $B={2,3}$, $1cdot B$ would be equal to the set ${2,3}$ is this set then by definition a left coset? Or is the set $acdot B,ainmathbb{N} = {2,3,4,6,8,9,12,....}$ a left coset?
– RM777
Nov 27 '18 at 19:01
For example let $cdot$ be the usual multiplication and $B={2,3}$, $1cdot B$ would be equal to the set ${2,3}$ is this set then by definition a left coset? Or is the set $acdot B,ainmathbb{N} = {2,3,4,6,8,9,12,....}$ a left coset?
– RM777
Nov 27 '18 at 19:01
add a comment |
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It's not very clear what you are asking: if $cdot$ is an operation on $Bbb{N}$, then $a cdot B subseteq Bbb{N}$ for any $a$ autormatically and $1 cdot Bbb{N}$ is a coset by definition. So what are you trying to prove?
– Rob Arthan
Nov 27 '18 at 18:47
For example let $cdot$ be the usual multiplication and $B={2,3}$, $1cdot B$ would be equal to the set ${2,3}$ is this set then by definition a left coset? Or is the set $acdot B,ainmathbb{N} = {2,3,4,6,8,9,12,....}$ a left coset?
– RM777
Nov 27 '18 at 19:01