Mixing
Induced Semigroup Structures via (left) Translation
Let $(M,circ)$ be any semigroup satisfying the following properties:
P-1: $text{For every } x,y,z in M text{, if } z circ x = z circ y , text{ then } , x = y$.
If $zeta in M$ we define $M_zeta = {m in M ; | ; m = zeta circ u}$.
We can define a binary operation $circ_zeta$ on $M_zeta$ as follows:
$tag 1 (zeta circ u) circ_zeta (zeta circ v) = zeta circ (u circ v)$
It is easy to see that $circ_zeta$ is an associative operation and that the semigroup $M_zeta$ also satisfies property $text{P-1}$.
I can think of several other properties of $M$ that would also hold true for $M_zeta$.
Is there any developed theory that analyzes these induced algebraic
structures?
Any answers that contain books/papers as well any results would be
helpful.
abstract-algebra reference-request soft-question book-recommendation semigroups
edited Nov 18 '18 at 2:56
Shaun
8,805113680
asked Nov 18 '18 at 2:35
CopyPasteIt
4,0451627
add a comment |
Let $(M,circ)$ be any semigroup satisfying the following properties:
P-1: $text{For every } x,y,z in M text{, if } z circ x = z circ y , text{ then } , x = y$.
If $zeta in M$ we define $M_zeta = {m in M ; | ; m = zeta circ u}$.
We can define a binary operation $circ_zeta$ on $M_zeta$ as follows:
$tag 1 (zeta circ u) circ_zeta (zeta circ v) = zeta circ (u circ v)$
It is easy to see that $circ_zeta$ is an associative operation and that the semigroup $M_zeta$ also satisfies property $text{P-1}$.
I can think of several other properties of $M$ that would also hold true for $M_zeta$.
Is there any developed theory that analyzes these induced algebraic
structures?
Any answers that contain books/papers as well any results would be
helpful.
abstract-algebra reference-request soft-question book-recommendation semigroups
edited Nov 18 '18 at 2:56
Shaun
8,805113680
asked Nov 18 '18 at 2:35
CopyPasteIt
4,0451627
add a comment |
Let $(M,circ)$ be any semigroup satisfying the following properties:
P-1: $text{For every } x,y,z in M text{, if } z circ x = z circ y , text{ then } , x = y$.
If $zeta in M$ we define $M_zeta = {m in M ; | ; m = zeta circ u}$.
We can define a binary operation $circ_zeta$ on $M_zeta$ as follows:
$tag 1 (zeta circ u) circ_zeta (zeta circ v) = zeta circ (u circ v)$
It is easy to see that $circ_zeta$ is an associative operation and that the semigroup $M_zeta$ also satisfies property $text{P-1}$.
I can think of several other properties of $M$ that would also hold true for $M_zeta$.
Is there any developed theory that analyzes these induced algebraic
structures?
Any answers that contain books/papers as well any results would be
helpful.
abstract-algebra reference-request soft-question book-recommendation semigroups
edited Nov 18 '18 at 2:56
Shaun
8,805113680
asked Nov 18 '18 at 2:35
CopyPasteIt
4,0451627
Let $(M,circ)$ be any semigroup satisfying the following properties:
P-1: $text{For every } x,y,z in M text{, if } z circ x = z circ y , text{ then } , x = y$.
If $zeta in M$ we define $M_zeta = {m in M ; | ; m = zeta circ u}$.
We can define a binary operation $circ_zeta$ on $M_zeta$ as follows:
$tag 1 (zeta circ u) circ_zeta (zeta circ v) = zeta circ (u circ v)$
It is easy to see that $circ_zeta$ is an associative operation and that the semigroup $M_zeta$ also satisfies property $text{P-1}$.
I can think of several other properties of $M$ that would also hold true for $M_zeta$.
Is there any developed theory that analyzes these induced algebraic
structures?
Any answers that contain books/papers as well any results would be
helpful.
abstract-algebra reference-request soft-question book-recommendation semigroups
abstract-algebra reference-request soft-question book-recommendation semigroups
edited Nov 18 '18 at 2:56
Shaun
8,805113680
asked Nov 18 '18 at 2:35
CopyPasteIt
4,0451627
edited Nov 18 '18 at 2:56
Shaun
8,805113680
asked Nov 18 '18 at 2:35
CopyPasteIt
4,0451627
edited Nov 18 '18 at 2:56
Shaun
8,805113680
edited Nov 18 '18 at 2:56
Shaun
8,805113680
edited Nov 18 '18 at 2:56
Shaun
8,805113680
8,805113680
asked Nov 18 '18 at 2:35
CopyPasteIt
4,0451627
asked Nov 18 '18 at 2:35
CopyPasteIt
4,0451627
asked Nov 18 '18 at 2:35
CopyPasteIt
4,0451627
4,0451627
add a comment |
add a comment |
1 Answer
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Since the map $umapstozetacirc u$ is an injection with range $M_zeta$, we actually get a bijection $Mto M_zeta$, and you just pulled over the semigroup operation of $M$.
Note that we could have done it with any other bijection $phi:Mto N$, by defining $acirc_N b:=phi^{-1}(a)circ_Mphi^{-1}(b)$.
Consequently, as semigroups, $M$ and $M_zeta$ are isomorphic.
answered Nov 21 '18 at 0:30
Berci
59.7k23672
Nevertheless, I've seen very similar and useful (!) ideas in e.g. functional analysis, defining a new inner product on the range of certain operators..
– Berci
Nov 21 '18 at 0:34
add a comment |
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1 Answer
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1 Answer
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active
oldest
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Since the map $umapstozetacirc u$ is an injection with range $M_zeta$, we actually get a bijection $Mto M_zeta$, and you just pulled over the semigroup operation of $M$.
Note that we could have done it with any other bijection $phi:Mto N$, by defining $acirc_N b:=phi^{-1}(a)circ_Mphi^{-1}(b)$.
Consequently, as semigroups, $M$ and $M_zeta$ are isomorphic.
answered Nov 21 '18 at 0:30
Berci
59.7k23672
Nevertheless, I've seen very similar and useful (!) ideas in e.g. functional analysis, defining a new inner product on the range of certain operators..
– Berci
Nov 21 '18 at 0:34
add a comment |
Since the map $umapstozetacirc u$ is an injection with range $M_zeta$, we actually get a bijection $Mto M_zeta$, and you just pulled over the semigroup operation of $M$.
Note that we could have done it with any other bijection $phi:Mto N$, by defining $acirc_N b:=phi^{-1}(a)circ_Mphi^{-1}(b)$.
Consequently, as semigroups, $M$ and $M_zeta$ are isomorphic.
answered Nov 21 '18 at 0:30
Berci
59.7k23672
Nevertheless, I've seen very similar and useful (!) ideas in e.g. functional analysis, defining a new inner product on the range of certain operators..
– Berci
Nov 21 '18 at 0:34
add a comment |
Since the map $umapstozetacirc u$ is an injection with range $M_zeta$, we actually get a bijection $Mto M_zeta$, and you just pulled over the semigroup operation of $M$.
Note that we could have done it with any other bijection $phi:Mto N$, by defining $acirc_N b:=phi^{-1}(a)circ_Mphi^{-1}(b)$.
Consequently, as semigroups, $M$ and $M_zeta$ are isomorphic.
answered Nov 21 '18 at 0:30
Berci
59.7k23672
Since the map $umapstozetacirc u$ is an injection with range $M_zeta$, we actually get a bijection $Mto M_zeta$, and you just pulled over the semigroup operation of $M$.
Note that we could have done it with any other bijection $phi:Mto N$, by defining $acirc_N b:=phi^{-1}(a)circ_Mphi^{-1}(b)$.
Consequently, as semigroups, $M$ and $M_zeta$ are isomorphic.
answered Nov 21 '18 at 0:30
Berci
59.7k23672
answered Nov 21 '18 at 0:30
Berci
59.7k23672
answered Nov 21 '18 at 0:30
Berci
59.7k23672
answered Nov 21 '18 at 0:30
Berci
59.7k23672
59.7k23672
Nevertheless, I've seen very similar and useful (!) ideas in e.g. functional analysis, defining a new inner product on the range of certain operators..
– Berci
Nov 21 '18 at 0:34
add a comment |
Nevertheless, I've seen very similar and useful (!) ideas in e.g. functional analysis, defining a new inner product on the range of certain operators..
– Berci
Nov 21 '18 at 0:34
Nevertheless, I've seen very similar and useful (!) ideas in e.g. functional analysis, defining a new inner product on the range of certain operators..
– Berci
Nov 21 '18 at 0:34
Nevertheless, I've seen very similar and useful (!) ideas in e.g. functional analysis, defining a new inner product on the range of certain operators..
– Berci
Nov 21 '18 at 0:34
add a comment |
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