$omega$th iteration of Cayley-Dickson construction
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The first nine Cayley-Dickson hypercomplex algebras are real (1D), complex (2D), quaternion (4D), octonion (8D), sedenion (16D), pathion (32D), chingon (64D), routon (128D), and the voudon (256D) -Source
The Cayley–Dickson construction can be carried on ad infinitum. -Wikipedia
What would the $omega$th step of this process produce?
In a surreal sense, would we then have Triernions, $epsilon$-ions, $omega$-ion, $pi$-ons, etc.?
Related:
Infinite-dimensional normed division algebras
Infinite-Dimensional Quadratic Forms Admitting Composition (pdf)
Non-Associative Algebras (pdf) Section 2.5 - Proof of Hurwit'z Theorem
sequences-and-series infinite-groups transfinite-recursion surreal-numbers transfinite-induction
asked Nov 11 at 4:33
meowzz
82212
add a comment |
up vote
-1
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favorite
The first nine Cayley-Dickson hypercomplex algebras are real (1D), complex (2D), quaternion (4D), octonion (8D), sedenion (16D), pathion (32D), chingon (64D), routon (128D), and the voudon (256D) -Source
The Cayley–Dickson construction can be carried on ad infinitum. -Wikipedia
What would the $omega$th step of this process produce?
In a surreal sense, would we then have Triernions, $epsilon$-ions, $omega$-ion, $pi$-ons, etc.?
Related:
Infinite-dimensional normed division algebras
Infinite-Dimensional Quadratic Forms Admitting Composition (pdf)
Non-Associative Algebras (pdf) Section 2.5 - Proof of Hurwit'z Theorem
sequences-and-series infinite-groups transfinite-recursion surreal-numbers transfinite-induction
asked Nov 11 at 4:33
meowzz
82212
2
So, after sedenions, we have pathions, chingons etc. How far up do we have to go to get klingons? Anyway, each embeds in the next one, so we can take a direct limit in order to get an $omega$-th term.
– Lord Shark the Unknown
Nov 11 at 6:01
1
@LordSharktheUnknown Since you got here first, do you want to expand your comment into an answer explaining the direct limit?
– Mark S.
Nov 11 at 19:10
You link a page explaining that "Triernions" don't exist, but ask if we would have them. If there is something about that page (or linked ones) you don't understand, I would recommend making that a separate question. Since the Surreals' $varepsilon$ and $pi$ aren't ordinals, I don't know what it would mean for something to have that many components.
– Mark S.
Nov 24 at 0:03
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
The first nine Cayley-Dickson hypercomplex algebras are real (1D), complex (2D), quaternion (4D), octonion (8D), sedenion (16D), pathion (32D), chingon (64D), routon (128D), and the voudon (256D) -Source
The Cayley–Dickson construction can be carried on ad infinitum. -Wikipedia
What would the $omega$th step of this process produce?
In a surreal sense, would we then have Triernions, $epsilon$-ions, $omega$-ion, $pi$-ons, etc.?
Related:
Infinite-dimensional normed division algebras
Infinite-Dimensional Quadratic Forms Admitting Composition (pdf)
Non-Associative Algebras (pdf) Section 2.5 - Proof of Hurwit'z Theorem
sequences-and-series infinite-groups transfinite-recursion surreal-numbers transfinite-induction
asked Nov 11 at 4:33
meowzz
82212
The first nine Cayley-Dickson hypercomplex algebras are real (1D), complex (2D), quaternion (4D), octonion (8D), sedenion (16D), pathion (32D), chingon (64D), routon (128D), and the voudon (256D) -Source
The Cayley–Dickson construction can be carried on ad infinitum. -Wikipedia
What would the $omega$th step of this process produce?
In a surreal sense, would we then have Triernions, $epsilon$-ions, $omega$-ion, $pi$-ons, etc.?
Related:
Infinite-dimensional normed division algebras
Infinite-Dimensional Quadratic Forms Admitting Composition (pdf)
Non-Associative Algebras (pdf) Section 2.5 - Proof of Hurwit'z Theorem
sequences-and-series infinite-groups transfinite-recursion surreal-numbers transfinite-induction
sequences-and-series infinite-groups transfinite-recursion surreal-numbers transfinite-induction
asked Nov 11 at 4:33
meowzz
82212
asked Nov 11 at 4:33
meowzz
82212
asked Nov 11 at 4:33
meowzz
82212
asked Nov 11 at 4:33
meowzz
82212
asked Nov 11 at 4:33
meowzz
82212
82212
2
So, after sedenions, we have pathions, chingons etc. How far up do we have to go to get klingons? Anyway, each embeds in the next one, so we can take a direct limit in order to get an $omega$-th term.
– Lord Shark the Unknown
Nov 11 at 6:01
1
@LordSharktheUnknown Since you got here first, do you want to expand your comment into an answer explaining the direct limit?
– Mark S.
Nov 11 at 19:10
You link a page explaining that "Triernions" don't exist, but ask if we would have them. If there is something about that page (or linked ones) you don't understand, I would recommend making that a separate question. Since the Surreals' $varepsilon$ and $pi$ aren't ordinals, I don't know what it would mean for something to have that many components.
– Mark S.
Nov 24 at 0:03
add a comment |
2
So, after sedenions, we have pathions, chingons etc. How far up do we have to go to get klingons? Anyway, each embeds in the next one, so we can take a direct limit in order to get an $omega$-th term.
– Lord Shark the Unknown
Nov 11 at 6:01
1
@LordSharktheUnknown Since you got here first, do you want to expand your comment into an answer explaining the direct limit?
– Mark S.
Nov 11 at 19:10
You link a page explaining that "Triernions" don't exist, but ask if we would have them. If there is something about that page (or linked ones) you don't understand, I would recommend making that a separate question. Since the Surreals' $varepsilon$ and $pi$ aren't ordinals, I don't know what it would mean for something to have that many components.
– Mark S.
Nov 24 at 0:03
2
2
So, after sedenions, we have pathions, chingons etc. How far up do we have to go to get klingons? Anyway, each embeds in the next one, so we can take a direct limit in order to get an $omega$-th term.
– Lord Shark the Unknown
Nov 11 at 6:01
So, after sedenions, we have pathions, chingons etc. How far up do we have to go to get klingons? Anyway, each embeds in the next one, so we can take a direct limit in order to get an $omega$-th term.
– Lord Shark the Unknown
Nov 11 at 6:01
1
1
@LordSharktheUnknown Since you got here first, do you want to expand your comment into an answer explaining the direct limit?
– Mark S.
Nov 11 at 19:10
@LordSharktheUnknown Since you got here first, do you want to expand your comment into an answer explaining the direct limit?
– Mark S.
Nov 11 at 19:10
You link a page explaining that "Triernions" don't exist, but ask if we would have them. If there is something about that page (or linked ones) you don't understand, I would recommend making that a separate question. Since the Surreals' $varepsilon$ and $pi$ aren't ordinals, I don't know what it would mean for something to have that many components.
– Mark S.
Nov 24 at 0:03
You link a page explaining that "Triernions" don't exist, but ask if we would have them. If there is something about that page (or linked ones) you don't understand, I would recommend making that a separate question. Since the Surreals' $varepsilon$ and $pi$ aren't ordinals, I don't know what it would mean for something to have that many components.
– Mark S.
Nov 24 at 0:03
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
As indicated in that diagram, each of these objects sits inside the bigger ones in a natural way. For instance, since octonions are pairs of quaternions, and quaternions are pairs of complex numbers, a complex number $(a,b)$ can be treated as an octonion by adding zeros: $(a,b)$ is like $(,((a,b),(0,0)),,((0,0),(0,0)),)$.
The Cayley-Dickson construction is set up in such a way that the arithmetic operations (addition and multiplication) are consistent with these sorts of identifications. E.g. if you multiply two complex numbers and then view the product as an octonion, you get the same answer as if you view the complex numbers as octonions first and then multiply.
Because of these identifications, a natural step $omega$ would be just to take the union of all of these sets of numbers, making identifications as needed. For example, when you need to test if a real number $r$ is equal to a complex number $(a,b)$, first view $r$ as $(r,0)$ and then test equality as complex numbers. Or if you need to add $(a,b)$ to $((c,d),(e,f))$, first view the complex number as the quaternion $((a,b),(0,0))$ and then add.
As Lord Shark the Unknown alluded to in a comment, whenever you have a system like this where "smaller" objects can be viewed as sitting inside larger ones by doing something (in this case: adding zeros) that's compatible with the structure (in this case: arithmetic operations), then you can collect everything together in something called the "direct limit" (in algebra). A slightly more abstract generalization of this idea would be the colimit in Category Theory.
answered Nov 24 at 0:16
Mark S.
11.4k22568
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
As indicated in that diagram, each of these objects sits inside the bigger ones in a natural way. For instance, since octonions are pairs of quaternions, and quaternions are pairs of complex numbers, a complex number $(a,b)$ can be treated as an octonion by adding zeros: $(a,b)$ is like $(,((a,b),(0,0)),,((0,0),(0,0)),)$.
The Cayley-Dickson construction is set up in such a way that the arithmetic operations (addition and multiplication) are consistent with these sorts of identifications. E.g. if you multiply two complex numbers and then view the product as an octonion, you get the same answer as if you view the complex numbers as octonions first and then multiply.
Because of these identifications, a natural step $omega$ would be just to take the union of all of these sets of numbers, making identifications as needed. For example, when you need to test if a real number $r$ is equal to a complex number $(a,b)$, first view $r$ as $(r,0)$ and then test equality as complex numbers. Or if you need to add $(a,b)$ to $((c,d),(e,f))$, first view the complex number as the quaternion $((a,b),(0,0))$ and then add.
As Lord Shark the Unknown alluded to in a comment, whenever you have a system like this where "smaller" objects can be viewed as sitting inside larger ones by doing something (in this case: adding zeros) that's compatible with the structure (in this case: arithmetic operations), then you can collect everything together in something called the "direct limit" (in algebra). A slightly more abstract generalization of this idea would be the colimit in Category Theory.
answered Nov 24 at 0:16
Mark S.
11.4k22568
add a comment |
up vote
1
down vote
accepted
As indicated in that diagram, each of these objects sits inside the bigger ones in a natural way. For instance, since octonions are pairs of quaternions, and quaternions are pairs of complex numbers, a complex number $(a,b)$ can be treated as an octonion by adding zeros: $(a,b)$ is like $(,((a,b),(0,0)),,((0,0),(0,0)),)$.
The Cayley-Dickson construction is set up in such a way that the arithmetic operations (addition and multiplication) are consistent with these sorts of identifications. E.g. if you multiply two complex numbers and then view the product as an octonion, you get the same answer as if you view the complex numbers as octonions first and then multiply.
Because of these identifications, a natural step $omega$ would be just to take the union of all of these sets of numbers, making identifications as needed. For example, when you need to test if a real number $r$ is equal to a complex number $(a,b)$, first view $r$ as $(r,0)$ and then test equality as complex numbers. Or if you need to add $(a,b)$ to $((c,d),(e,f))$, first view the complex number as the quaternion $((a,b),(0,0))$ and then add.
As Lord Shark the Unknown alluded to in a comment, whenever you have a system like this where "smaller" objects can be viewed as sitting inside larger ones by doing something (in this case: adding zeros) that's compatible with the structure (in this case: arithmetic operations), then you can collect everything together in something called the "direct limit" (in algebra). A slightly more abstract generalization of this idea would be the colimit in Category Theory.
answered Nov 24 at 0:16
Mark S.
11.4k22568
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
As indicated in that diagram, each of these objects sits inside the bigger ones in a natural way. For instance, since octonions are pairs of quaternions, and quaternions are pairs of complex numbers, a complex number $(a,b)$ can be treated as an octonion by adding zeros: $(a,b)$ is like $(,((a,b),(0,0)),,((0,0),(0,0)),)$.
The Cayley-Dickson construction is set up in such a way that the arithmetic operations (addition and multiplication) are consistent with these sorts of identifications. E.g. if you multiply two complex numbers and then view the product as an octonion, you get the same answer as if you view the complex numbers as octonions first and then multiply.
Because of these identifications, a natural step $omega$ would be just to take the union of all of these sets of numbers, making identifications as needed. For example, when you need to test if a real number $r$ is equal to a complex number $(a,b)$, first view $r$ as $(r,0)$ and then test equality as complex numbers. Or if you need to add $(a,b)$ to $((c,d),(e,f))$, first view the complex number as the quaternion $((a,b),(0,0))$ and then add.
As Lord Shark the Unknown alluded to in a comment, whenever you have a system like this where "smaller" objects can be viewed as sitting inside larger ones by doing something (in this case: adding zeros) that's compatible with the structure (in this case: arithmetic operations), then you can collect everything together in something called the "direct limit" (in algebra). A slightly more abstract generalization of this idea would be the colimit in Category Theory.
answered Nov 24 at 0:16
Mark S.
11.4k22568
As indicated in that diagram, each of these objects sits inside the bigger ones in a natural way. For instance, since octonions are pairs of quaternions, and quaternions are pairs of complex numbers, a complex number $(a,b)$ can be treated as an octonion by adding zeros: $(a,b)$ is like $(,((a,b),(0,0)),,((0,0),(0,0)),)$.
The Cayley-Dickson construction is set up in such a way that the arithmetic operations (addition and multiplication) are consistent with these sorts of identifications. E.g. if you multiply two complex numbers and then view the product as an octonion, you get the same answer as if you view the complex numbers as octonions first and then multiply.
Because of these identifications, a natural step $omega$ would be just to take the union of all of these sets of numbers, making identifications as needed. For example, when you need to test if a real number $r$ is equal to a complex number $(a,b)$, first view $r$ as $(r,0)$ and then test equality as complex numbers. Or if you need to add $(a,b)$ to $((c,d),(e,f))$, first view the complex number as the quaternion $((a,b),(0,0))$ and then add.
As Lord Shark the Unknown alluded to in a comment, whenever you have a system like this where "smaller" objects can be viewed as sitting inside larger ones by doing something (in this case: adding zeros) that's compatible with the structure (in this case: arithmetic operations), then you can collect everything together in something called the "direct limit" (in algebra). A slightly more abstract generalization of this idea would be the colimit in Category Theory.
answered Nov 24 at 0:16
Mark S.
11.4k22568
answered Nov 24 at 0:16
Mark S.
11.4k22568
answered Nov 24 at 0:16
Mark S.
11.4k22568
answered Nov 24 at 0:16
Mark S.
11.4k22568
11.4k22568
add a comment |
add a comment |
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So, after sedenions, we have pathions, chingons etc. How far up do we have to go to get klingons? Anyway, each embeds in the next one, so we can take a direct limit in order to get an $omega$-th term.
– Lord Shark the Unknown
Nov 11 at 6:01
1
@LordSharktheUnknown Since you got here first, do you want to expand your comment into an answer explaining the direct limit?
– Mark S.
Nov 11 at 19:10
You link a page explaining that "Triernions" don't exist, but ask if we would have them. If there is something about that page (or linked ones) you don't understand, I would recommend making that a separate question. Since the Surreals' $varepsilon$ and $pi$ aren't ordinals, I don't know what it would mean for something to have that many components.
– Mark S.
Nov 24 at 0:03