05/14/2021, 06:42 PM
Thank you for the time James. I apologize for some grammar errors.
I'm sorry but here you lose me. In which local framework elements of [f,g] can be thought as part of a conjugacy class in the literal meaning of the term?
I call two elements f,g related by conjugation (conjugated) if exists, at least, a third object x s.t. xf=gx. In the case of \( {\rm Rsd}(f,\zeta) \) we have that each pair is conjugated by a path, or more then one, (by your compositional-Cauchy thm (CCT)). Now, if there exists more objects x,y,z that conjugate f and g, i.e.
xf=gx, yf=gy adn zf=gz
we can say that x,y,z are three different ways in which f and g are in relation (conjugated). We can see x,y,z as modes of being conjugated of f and g. Using the geometric intuition of curves and your CCT we can see x,y,z as oriented paths joining f to g.
If we want to retain the conceptual connection with conjugacy, which is fine to me, I find appropriate to call [f,g] the set of modes of f to being conjugated to g.
Some alternatives:
Are the sets [f,g] equivalence classes of some equivalence relation?
Nope, because they are never disjoint as far as gorups, monoids and semigroups are concerned. If we collect all the [f,g] they do not form a partition of the total set of functions (to prove this just consider [id,id] everything commutes with the identity).
It is an open problem if elements of [f,g] are conjugated and how to classify them. Let x,y such that xf=gx and yf=gy... When is there an object s such that sx=ys?
What do you mean?
I'm not sure. Change of coordinates system?
I was thinking of change of time: because an element x conjugating f to g changes the frame of reference in some sense... (that's the case of linear algebra) but since we are talking about two iterations maybe we could force the analogy by claiming that x is translating the time of f to the time of g.
This can be confusing if we think of changing time as changing our monoid of time, e.g. when we extend from N to Z, or from integer iteration (Z) to complex iteration ©.
But I've a proof (that has to do with rank=1 functions) that the change of time monoid is related to some extent to the idea that \( \chi \in [f,g] \) is really changing the time from f to the time g.
Yes. As I was saying, the relation of conjugacy, as I defined it, is an equivalence only for groups. In monoids and semigroups the possible definitions of conjugacy become dependent on orientation and on other technical difficulties. In other words the concept fragments in a couple of variants. In "the literature", as far as I know and remember that I'm just a nobody that didn't get past the first semester of university, they still call them conjugacy classes. Usually, all the different possible definitions are equivalent when we restrict ourselves to groups.
The two most striking ones, that hold universally in semigroups and monoids (i.e semigroups with identity element), are the relations of isomorphism and of connectedness. These two relations define classes that are called isomorphism classes and, respectively, connected components. (note: this is not fringe math but very well know. See Tom Leinster on MO.)
As a side note: I'd love to hear opinions of historical heavyweights of the forum like Trappmann, Robbins, Sheldonison, Tommy, Gottfried and Mick
(05/14/2021, 03:36 PM)JmsNxn Wrote: Personally, I'm a fan of conjugate classes; but you're right; technically that's incorrect. I would usually always consider invertible (at least, locally) maps. So, in that frame work it's correct.
I'm sorry but here you lose me. In which local framework elements of [f,g] can be thought as part of a conjugacy class in the literal meaning of the term?
I call two elements f,g related by conjugation (conjugated) if exists, at least, a third object x s.t. xf=gx. In the case of \( {\rm Rsd}(f,\zeta) \) we have that each pair is conjugated by a path, or more then one, (by your compositional-Cauchy thm (CCT)). Now, if there exists more objects x,y,z that conjugate f and g, i.e.
xf=gx, yf=gy adn zf=gz
we can say that x,y,z are three different ways in which f and g are in relation (conjugated). We can see x,y,z as modes of being conjugated of f and g. Using the geometric intuition of curves and your CCT we can see x,y,z as oriented paths joining f to g.
If we want to retain the conceptual connection with conjugacy, which is fine to me, I find appropriate to call [f,g] the set of modes of f to being conjugated to g.
Some alternatives:
- [f,g] is the set of oriented paths from f to g
- [f,g] is the set of transformations of f into g
- [f,g] is the set of f-iterations of g or of f-superfunctions of g
- [f,g] is the set of g-logarithms of f or of g-abel functions of f
Are the sets [f,g] equivalence classes of some equivalence relation?
Nope, because they are never disjoint as far as gorups, monoids and semigroups are concerned. If we collect all the [f,g] they do not form a partition of the total set of functions (to prove this just consider [id,id] everything commutes with the identity).
It is an open problem if elements of [f,g] are conjugated and how to classify them. Let x,y such that xf=gx and yf=gy... When is there an object s such that sx=ys?
Quote: I do like equivariant, but that seems a tad restrictive when we're just considering one variable.
What do you mean?
Quote:Change of base honestly sounds like the best one, if you could only make this a better term than just change of base. It also fits more at home with this forum, where it is an equivalent to what we typically call change of base.
I'm not sure. Change of coordinates system?
I was thinking of change of time: because an element x conjugating f to g changes the frame of reference in some sense... (that's the case of linear algebra) but since we are talking about two iterations maybe we could force the analogy by claiming that x is translating the time of f to the time of g.
This can be confusing if we think of changing time as changing our monoid of time, e.g. when we extend from N to Z, or from integer iteration (Z) to complex iteration ©.
But I've a proof (that has to do with rank=1 functions) that the change of time monoid is related to some extent to the idea that \( \chi \in [f,g] \) is really changing the time from f to the time g.
Quote:I wonder if there is a term for conjugate classes in monoids? Or at least an equivalent for monoids?
Yes. As I was saying, the relation of conjugacy, as I defined it, is an equivalence only for groups. In monoids and semigroups the possible definitions of conjugacy become dependent on orientation and on other technical difficulties. In other words the concept fragments in a couple of variants. In "the literature", as far as I know and remember that I'm just a nobody that didn't get past the first semester of university, they still call them conjugacy classes. Usually, all the different possible definitions are equivalent when we restrict ourselves to groups.
The two most striking ones, that hold universally in semigroups and monoids (i.e semigroups with identity element), are the relations of isomorphism and of connectedness. These two relations define classes that are called isomorphism classes and, respectively, connected components. (note: this is not fringe math but very well know. See Tom Leinster on MO.)
As a side note: I'd love to hear opinions of historical heavyweights of the forum like Trappmann, Robbins, Sheldonison, Tommy, Gottfried and Mick
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)