05/06/2021, 11:54 AM
(This post was last modified: 05/06/2021, 07:30 PM by MphLee.
Edit Reason: GREATLY IMPROVED
)
BIG EDIT
Hi, I'm sad that I have to answer briefly. I'm taking some notes and comments about the key parts of this thread and I'll come up with a complete and polished opinion ASAP.
Lately, I got virtually 0 time to put into the forum.
Anyways, I can give my few cents.
About the notation
I find the notation you propose over-complicated. Those are just sets and we don't need to write in front of that a representative of that set. Instead of \( \zeta\{f,g\} \) that conceptually invites you to see it as a function. You should write \( \zeta\in \{f,g\} \) instead, hinting at the fact that it is a multi-valued function. If one finds a way to systematically pick from every set a special solution, we can make a new functional notation like \( \{f,g\}f=g\{f,g\} \) or \( (f\oplus g)f=g(f\oplus g) \).
Since the beginning of my path (2012-2013) I always have regarded having to do with a set of multiple possible solutions as a limitation. I wanted desperately to find THE special solution in every set but as JmsNxn states "The really hard part, is distinguishing between each member of the set". For that reason I started acting as if that set was a unique function and I derived the same "laws" you derived. I used the notation \( f\oplus g \) naming it "Abel sum".(*)
After 10 years I can claim that this stubbornness slowed me down! In fact I understood that we have to embrace that multiplicity and see what happens. What happens is category theory!
One of my old dreams is to come up with an abstract classification of those sets and with an inner classification of the solutions inside those sets... I guess the way to go is to look at the theory of bundles and connections but is tremendously hard (It links us to mathematical physics.).
Nowadays (v) I'd just write \( [f,g] \). When the meaning is not clear because our functions come from different monoids \( f,g\in M \) and \( h,l\in L \) of functions, I'd go for \( M(f,g) \) and \( L(h,l) \). It turns out that those sets are nothing but Hom-sets and they define a functor from the category of dynamical system (or N-actions) to the category of sets and functions.
One other example of why the notation can be not enough comfortable is the following: I find more gentle for who writes to denote with f(x) not the function f, but the value of the function f at x. So when you write \( \alpha\{f(z)\}(z) \) we should better have \( \alpha\{f\}(z) \) where \( \alpha\{f\} \) is the abel function of f. The consequence of not doing that is that, on a bad day, you could end up writing things like
\( g(\zeta\{f(z);g(z)\}(z^2+6))=\zeta\{f(z^3)z^5;g(z)\}(f(z))
\)
This can seem like pedantic and useless. I don't know what happens in the analysis realm, but in set theory and in algebra finding the right notation open the doors to new intuition and abstraction (+old and seemingly unrelated theorems become available.).
About the Laws
Those sets \( \zeta\{f,g\} \) you are defining satisfy indeed the properties you describe in the opening post. Those properties have nothing to do with the fact that our functions are analytic, continuous or multi-valued. Those "laws", in fact do not even derive from f,g and \( \zeta \) being functions. Most of those propositions come from free from the fact functions are elements of a monoid, wrt composition. Those "laws" are low-level shadows of a more vast zoo of "elementary laws" that come from the structure of categories(iv) !
Some of them do not even require a monoid structure to be deduced: those are a much more "primordial" kind of laws that hold, for example, also for solution sets of equations in more general cases. To take as an example the way every solution set of a non-homogeneous system of linear equations (the fiber/preimage of a linear transformation at a given vector) can be decomposed/factored into a traslation of a solution set of an homogeneous system (you can know all the solutions of Ax=b if you know a single solution and the solution set of Ax=0, i.e. the null space of the matrix). In the same way, the set of antiderivative of a function can be decomposed into a sum of a known solution with the set of antiderivatives of the zero (aka constants +C).
Just some examples:
The analogy with basis and eigendecomposition is not that clear to me. I'm working on at least three different views on that but still can not come up with an unifying vision. For sure finding the superfunction amounts to a change the base in some sense. Amounts to having a dynamical system and changing the base to a new base where the dynamics of the initial map become "linear" (see Schroeder and Abel). I'd like to know if you have more insight into this.
Oh, btw,
welcome to the forum!
(*) The additive notation, that I don't support as strongly anymore was chosen to make evident the fact that the operation has a left and right inverse operation (conjugation) and was a pivotal for the intuition of hyperoperation rank-as-iteration of the"Abel sum". \( f\oplus (f\oplus (...\oplus f))={}^{s}f \). I wanted so badly to make formal JmsNxn's diamond operator.
(**) Impossible if the monoid is non-trivial: if it is not then every element is conjugated to itself by identity AND itself. It is impossible even given the most permissive interpretation of the condition: if f and g ARE NOT idempotent. A solution if x solves xf=gx then xf and xf^2 are new solutions.
(***) Anticommutativity-like properties showing up is an interesting thing because it hints at some links with Lie brackets or metric spaces.
(iv) For a crash intro into categories (for the algebraic-discouraged-teens and not anymore-teens) see this post
and this soft intro to the general role of categories.pdf.
(v) With "nowadays" I can send you to my recent post about generalized superfunction tricks. But that version is old and I worked on it a lot. I give you instead the new section 1.1.2 on notation from that paper. Ii is still work in progress.
On notation.pdf (Size: 256.67 KB / Downloads: 833)
Hi, I'm sad that I have to answer briefly. I'm taking some notes and comments about the key parts of this thread and I'll come up with a complete and polished opinion ASAP.
Lately, I got virtually 0 time to put into the forum.
Anyways, I can give my few cents.
About the notation
I find the notation you propose over-complicated. Those are just sets and we don't need to write in front of that a representative of that set. Instead of \( \zeta\{f,g\} \) that conceptually invites you to see it as a function. You should write \( \zeta\in \{f,g\} \) instead, hinting at the fact that it is a multi-valued function. If one finds a way to systematically pick from every set a special solution, we can make a new functional notation like \( \{f,g\}f=g\{f,g\} \) or \( (f\oplus g)f=g(f\oplus g) \).
Since the beginning of my path (2012-2013) I always have regarded having to do with a set of multiple possible solutions as a limitation. I wanted desperately to find THE special solution in every set but as JmsNxn states "The really hard part, is distinguishing between each member of the set". For that reason I started acting as if that set was a unique function and I derived the same "laws" you derived. I used the notation \( f\oplus g \) naming it "Abel sum".(*)
After 10 years I can claim that this stubbornness slowed me down! In fact I understood that we have to embrace that multiplicity and see what happens. What happens is category theory!
One of my old dreams is to come up with an abstract classification of those sets and with an inner classification of the solutions inside those sets... I guess the way to go is to look at the theory of bundles and connections but is tremendously hard (It links us to mathematical physics.).
Nowadays (v) I'd just write \( [f,g] \). When the meaning is not clear because our functions come from different monoids \( f,g\in M \) and \( h,l\in L \) of functions, I'd go for \( M(f,g) \) and \( L(h,l) \). It turns out that those sets are nothing but Hom-sets and they define a functor from the category of dynamical system (or N-actions) to the category of sets and functions.
One other example of why the notation can be not enough comfortable is the following: I find more gentle for who writes to denote with f(x) not the function f, but the value of the function f at x. So when you write \( \alpha\{f(z)\}(z) \) we should better have \( \alpha\{f\}(z) \) where \( \alpha\{f\} \) is the abel function of f. The consequence of not doing that is that, on a bad day, you could end up writing things like
\( g(\zeta\{f(z);g(z)\}(z^2+6))=\zeta\{f(z^3)z^5;g(z)\}(f(z))
\)
This can seem like pedantic and useless. I don't know what happens in the analysis realm, but in set theory and in algebra finding the right notation open the doors to new intuition and abstraction (+old and seemingly unrelated theorems become available.).
About the Laws
Those sets \( \zeta\{f,g\} \) you are defining satisfy indeed the properties you describe in the opening post. Those properties have nothing to do with the fact that our functions are analytic, continuous or multi-valued. Those "laws", in fact do not even derive from f,g and \( \zeta \) being functions. Most of those propositions come from free from the fact functions are elements of a monoid, wrt composition. Those "laws" are low-level shadows of a more vast zoo of "elementary laws" that come from the structure of categories(iv) !
Some of them do not even require a monoid structure to be deduced: those are a much more "primordial" kind of laws that hold, for example, also for solution sets of equations in more general cases. To take as an example the way every solution set of a non-homogeneous system of linear equations (the fiber/preimage of a linear transformation at a given vector) can be decomposed/factored into a traslation of a solution set of an homogeneous system (you can know all the solutions of Ax=b if you know a single solution and the solution set of Ax=0, i.e. the null space of the matrix). In the same way, the set of antiderivative of a function can be decomposed into a sum of a known solution with the set of antiderivatives of the zero (aka constants +C).
Just some examples:
- if every solution set had a unique element(**) then "Inversion Law" would be named anticommutativity(***) and is equivalent to conjugacy being symmetric. It does hold exclusively for groups of functions and it's basically a corollary of working with a groupoid. \( {\rm Hom}(g,f)={\rm Hom}(f,g)^{-1} \)
- The "cancel law" is basically composition in the category or, from another point of view, transitivity of the conjugation.
\( \circ :{\rm Hom}(g,h)\times {\rm Hom}(f,g)\to{\rm Hom}(f,h) \)
- Following the properties of equivalence relations there is the las one, i.e. reflexivity, and we can exhibit it as the third natural property of those sets: identity is a solution of fx=xf
\( {\rm id}_X=f^0 \in {\rm Hom}(f,f) \)
- Law 5 comes from the solution set being a torsor (like solution sets of non-homogeneous linear equations are affine spaces). But it is also a corollary of the cancel law.
\( \circ :{\rm Hom}(g,g)\times {\rm Hom}(f,g)\to{\rm Hom}(f,g) \)
- The invariant law is more interesting and subtle. I still don't know where it comes from exactly; I'm currently testing how much one can generalize it.
- The thins about fixed point is standard knowledge ofc. Fixed points are dynamical properties. Two dynamics being conjugated. Every \( \chi\in [f,g] \), i.e. is a solution to xf=gx is a morphism of dynamical system. Morphism of dynamical system preserve positive dynamical properties (e.g having n-periodic points) and reflect negative properties. [See the book Conspetual Mathematics by Lawver-Schanuel]
The analogy with basis and eigendecomposition is not that clear to me. I'm working on at least three different views on that but still can not come up with an unifying vision. For sure finding the superfunction amounts to a change the base in some sense. Amounts to having a dynamical system and changing the base to a new base where the dynamics of the initial map become "linear" (see Schroeder and Abel). I'd like to know if you have more insight into this.
Oh, btw,
welcome to the forum!(*) The additive notation, that I don't support as strongly anymore was chosen to make evident the fact that the operation has a left and right inverse operation (conjugation) and was a pivotal for the intuition of hyperoperation rank-as-iteration of the"Abel sum". \( f\oplus (f\oplus (...\oplus f))={}^{s}f \). I wanted so badly to make formal JmsNxn's diamond operator.
(**) Impossible if the monoid is non-trivial: if it is not then every element is conjugated to itself by identity AND itself. It is impossible even given the most permissive interpretation of the condition: if f and g ARE NOT idempotent. A solution if x solves xf=gx then xf and xf^2 are new solutions.
(***) Anticommutativity-like properties showing up is an interesting thing because it hints at some links with Lie brackets or metric spaces.
(iv) For a crash intro into categories (for the algebraic-discouraged-teens and not anymore-teens) see this post
Composition, bullet notation and the general role of categories, (February 2, 2021)
and this soft intro to the general role of categories.pdf.
(v) With "nowadays" I can send you to my recent post about generalized superfunction tricks. But that version is old and I worked on it a lot. I give you instead the new section 1.1.2 on notation from that paper. Ii is still work in progress.
On notation.pdf (Size: 256.67 KB / Downloads: 833)
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)