Mixing
Neutral element for a Cantor Set.
On Wiki I found the following statement:
$T_L$ and $T_R$ together with function composition forms a monoid.
I am able to prove the associativity of the composition operation, but what will be a neutral element for the operation?
Thank you.
monoid cantor-set
asked Nov 22 '18 at 1:43
yourbuddyyourbuddy
212
add a comment |
On Wiki I found the following statement:
$T_L$ and $T_R$ together with function composition forms a monoid.
I am able to prove the associativity of the composition operation, but what will be a neutral element for the operation?
Thank you.
monoid cantor-set
asked Nov 22 '18 at 1:43
yourbuddyyourbuddy
212
2
It shouldn't say forms a monoid, it should say generates a monoid. In particular the identity element is the "empty composition", i.e. the identity function on the Cantor set.
– Mees de Vries
Nov 22 '18 at 1:48
@MeesdeVries Or, the empty composition. :P
– Noah Schweber
Nov 22 '18 at 3:03
add a comment |
On Wiki I found the following statement:
$T_L$ and $T_R$ together with function composition forms a monoid.
I am able to prove the associativity of the composition operation, but what will be a neutral element for the operation?
Thank you.
monoid cantor-set
asked Nov 22 '18 at 1:43
yourbuddyyourbuddy
212
On Wiki I found the following statement:
$T_L$ and $T_R$ together with function composition forms a monoid.
I am able to prove the associativity of the composition operation, but what will be a neutral element for the operation?
Thank you.
monoid cantor-set
monoid cantor-set
asked Nov 22 '18 at 1:43
yourbuddyyourbuddy
212
asked Nov 22 '18 at 1:43
yourbuddyyourbuddy
212
asked Nov 22 '18 at 1:43
yourbuddyyourbuddy
212
asked Nov 22 '18 at 1:43
yourbuddyyourbuddy
212
asked Nov 22 '18 at 1:43
yourbuddyyourbuddy
212
212
2
It shouldn't say forms a monoid, it should say generates a monoid. In particular the identity element is the "empty composition", i.e. the identity function on the Cantor set.
– Mees de Vries
Nov 22 '18 at 1:48
@MeesdeVries Or, the empty composition. :P
– Noah Schweber
Nov 22 '18 at 3:03
add a comment |
2
It shouldn't say forms a monoid, it should say generates a monoid. In particular the identity element is the "empty composition", i.e. the identity function on the Cantor set.
– Mees de Vries
Nov 22 '18 at 1:48
@MeesdeVries Or, the empty composition. :P
– Noah Schweber
Nov 22 '18 at 3:03
2
2
It shouldn't say forms a monoid, it should say generates a monoid. In particular the identity element is the "empty composition", i.e. the identity function on the Cantor set.
– Mees de Vries
Nov 22 '18 at 1:48
It shouldn't say forms a monoid, it should say generates a monoid. In particular the identity element is the "empty composition", i.e. the identity function on the Cantor set.
– Mees de Vries
Nov 22 '18 at 1:48
@MeesdeVries Or, the empty composition. :P
– Noah Schweber
Nov 22 '18 at 3:03
@MeesdeVries Or, the empty composition. :P
– Noah Schweber
Nov 22 '18 at 3:03
add a comment |
1 Answer
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As Mees de Vries says, it should really say "generates" rather than "forms," and you're exactly right about why: there's no identity (= neutral) element. However, if we broaden our horizons a bit, it can be true as written! Similarly to how the empty sum, empty product,and various other "empty operations" can be defined, there is a natural notion of "composition of no functions" - namely, the empty composition is the identity function, and this accounts for the "missing" identity element in our monoid here.
Let's think about a single function first. Composition satisfies the additivity rule $f^mcirc f^n=f^{m+n}$ for $m,n$ natural numbers $>0$, where "$f^k$" denotes the function gotten by composing $f$ with itself $k$ times. Based on this there is a unique way to define $f^0$ so as to follow the pattern: $f^0$ should be the identity map.
Now you're looking at a slightly more complicated situation: you have two functions involved, and can compose them with each other. Here natural numbers $>0$ get replaced by finite binary strings of length $>0$: e.g. $$T_LT_RT_L$$ is represented by the string $010$ (arbitrarily setting $L=0,R=1$). In place of additivity, we have a concatenation rule: the composition of the functions represented by the strings $sigma$ and $tau$ is the function represented by the string $sigmatau$. Now, what should the function corresponding to the empty string be, based on this rule?
answered Nov 22 '18 at 3:10
Noah SchweberNoah Schweber
122k10149284
add a comment |
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As Mees de Vries says, it should really say "generates" rather than "forms," and you're exactly right about why: there's no identity (= neutral) element. However, if we broaden our horizons a bit, it can be true as written! Similarly to how the empty sum, empty product,and various other "empty operations" can be defined, there is a natural notion of "composition of no functions" - namely, the empty composition is the identity function, and this accounts for the "missing" identity element in our monoid here.
Let's think about a single function first. Composition satisfies the additivity rule $f^mcirc f^n=f^{m+n}$ for $m,n$ natural numbers $>0$, where "$f^k$" denotes the function gotten by composing $f$ with itself $k$ times. Based on this there is a unique way to define $f^0$ so as to follow the pattern: $f^0$ should be the identity map.
Now you're looking at a slightly more complicated situation: you have two functions involved, and can compose them with each other. Here natural numbers $>0$ get replaced by finite binary strings of length $>0$: e.g. $$T_LT_RT_L$$ is represented by the string $010$ (arbitrarily setting $L=0,R=1$). In place of additivity, we have a concatenation rule: the composition of the functions represented by the strings $sigma$ and $tau$ is the function represented by the string $sigmatau$. Now, what should the function corresponding to the empty string be, based on this rule?
answered Nov 22 '18 at 3:10
Noah SchweberNoah Schweber
122k10149284
add a comment |
As Mees de Vries says, it should really say "generates" rather than "forms," and you're exactly right about why: there's no identity (= neutral) element. However, if we broaden our horizons a bit, it can be true as written! Similarly to how the empty sum, empty product,and various other "empty operations" can be defined, there is a natural notion of "composition of no functions" - namely, the empty composition is the identity function, and this accounts for the "missing" identity element in our monoid here.
Let's think about a single function first. Composition satisfies the additivity rule $f^mcirc f^n=f^{m+n}$ for $m,n$ natural numbers $>0$, where "$f^k$" denotes the function gotten by composing $f$ with itself $k$ times. Based on this there is a unique way to define $f^0$ so as to follow the pattern: $f^0$ should be the identity map.
Now you're looking at a slightly more complicated situation: you have two functions involved, and can compose them with each other. Here natural numbers $>0$ get replaced by finite binary strings of length $>0$: e.g. $$T_LT_RT_L$$ is represented by the string $010$ (arbitrarily setting $L=0,R=1$). In place of additivity, we have a concatenation rule: the composition of the functions represented by the strings $sigma$ and $tau$ is the function represented by the string $sigmatau$. Now, what should the function corresponding to the empty string be, based on this rule?
answered Nov 22 '18 at 3:10
Noah SchweberNoah Schweber
122k10149284
add a comment |
As Mees de Vries says, it should really say "generates" rather than "forms," and you're exactly right about why: there's no identity (= neutral) element. However, if we broaden our horizons a bit, it can be true as written! Similarly to how the empty sum, empty product,and various other "empty operations" can be defined, there is a natural notion of "composition of no functions" - namely, the empty composition is the identity function, and this accounts for the "missing" identity element in our monoid here.
Let's think about a single function first. Composition satisfies the additivity rule $f^mcirc f^n=f^{m+n}$ for $m,n$ natural numbers $>0$, where "$f^k$" denotes the function gotten by composing $f$ with itself $k$ times. Based on this there is a unique way to define $f^0$ so as to follow the pattern: $f^0$ should be the identity map.
Now you're looking at a slightly more complicated situation: you have two functions involved, and can compose them with each other. Here natural numbers $>0$ get replaced by finite binary strings of length $>0$: e.g. $$T_LT_RT_L$$ is represented by the string $010$ (arbitrarily setting $L=0,R=1$). In place of additivity, we have a concatenation rule: the composition of the functions represented by the strings $sigma$ and $tau$ is the function represented by the string $sigmatau$. Now, what should the function corresponding to the empty string be, based on this rule?
answered Nov 22 '18 at 3:10
Noah SchweberNoah Schweber
122k10149284
As Mees de Vries says, it should really say "generates" rather than "forms," and you're exactly right about why: there's no identity (= neutral) element. However, if we broaden our horizons a bit, it can be true as written! Similarly to how the empty sum, empty product,and various other "empty operations" can be defined, there is a natural notion of "composition of no functions" - namely, the empty composition is the identity function, and this accounts for the "missing" identity element in our monoid here.
Let's think about a single function first. Composition satisfies the additivity rule $f^mcirc f^n=f^{m+n}$ for $m,n$ natural numbers $>0$, where "$f^k$" denotes the function gotten by composing $f$ with itself $k$ times. Based on this there is a unique way to define $f^0$ so as to follow the pattern: $f^0$ should be the identity map.
Now you're looking at a slightly more complicated situation: you have two functions involved, and can compose them with each other. Here natural numbers $>0$ get replaced by finite binary strings of length $>0$: e.g. $$T_LT_RT_L$$ is represented by the string $010$ (arbitrarily setting $L=0,R=1$). In place of additivity, we have a concatenation rule: the composition of the functions represented by the strings $sigma$ and $tau$ is the function represented by the string $sigmatau$. Now, what should the function corresponding to the empty string be, based on this rule?
answered Nov 22 '18 at 3:10
Noah SchweberNoah Schweber
122k10149284
edited Nov 22 '18 at 3:18
answered Nov 22 '18 at 3:10
Noah SchweberNoah Schweber
122k10149284
answered Nov 22 '18 at 3:10
Noah SchweberNoah Schweber
122k10149284
answered Nov 22 '18 at 3:10
Noah SchweberNoah Schweber
122k10149284
122k10149284
add a comment |
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It shouldn't say forms a monoid, it should say generates a monoid. In particular the identity element is the "empty composition", i.e. the identity function on the Cantor set.
– Mees de Vries
Nov 22 '18 at 1:48
@MeesdeVries Or, the empty composition. :P
– Noah Schweber
Nov 22 '18 at 3:03