Yes when the index is real, I've only every looked at the riemann-stieljtes construction. Consider a partition \( \mathcal{P} \) of \( [a,b] \), lets call it \( s_{j+1} \le s_j^* \le s_j \) where \( b = s_0 > s_1 > s_2 >...>s_n = a \)--where here \( s_j^* \) is our sample point. Note I've written the partition as descending. Then, we call the partial compositions,
\(
Y_{\mathcal{P}}(z) = \Omega_{j=0}^{n-1} z + \phi(s_j^*,z)(s_j - s_{j+1})\bullet z\\
\)
Where we assume \( \phi(s,z) \) is some nice function. So as we limit the partition (as it gets finer and finer)--namely \( s_{j+1} - s_j \to 0 \)--under well enough conditions (holomorphy suffices) we get,
\(
\lim_{||\mathcal{P}||\to 0} Y_{\mathcal{P}}(z) = \int_{a}^b \phi(s,z)\,ds\bullet z\\
\)
Now, what this notation means, because we get two things for the price of one, is if,
\(
y(x) = \int_a^x \phi(s,z)\,ds\bullet z\\
\)
Then,
\(
y(a) = z\,\,\,\text{and}\,\,\, y'(x) = \phi(x,y(x))\\
\)
Which is just the equation of a first order differential equation. These motions date allllllllllllll the way back to Euler (usually the bastardization we see is Euler's method, but it's far more advanced). And it's actually kind of funny how much compositional analysis Euler used, which seems to have been just,, well, forgotten... This also helps us learn where the singularities arise--they are where the differential equation blows up.
For instance, take \( \phi(s,z) = z^2 \) then,
\(
\int_a^b z^2 \,ds\bullet z = \frac{1}{\frac{1}{z} + a -b}\\
\)
So where ever \( a-b = \frac{1}{z} \) we know instantly that this infinite composition can't converge because the differential equation blows up. This is really the best I could do--but I did prove some general normality conditions on,
\(
\Omega_{j=0}^{n-1} h_{jn}(s,z)\,\bullet z\\
\)
As we let \( n\to\infty \) then \( h_{jn}(s,z) \) needs to behave either discretely (the infinite composition way) or continuously (which looks something like the riemann-stieljtes composition, but not necessarily).
EDIT:
As to doing this with monoids. I can't even imagine. Maybe something like Tate's thesis with adeles and nonsense. You can count me out for that one, lmao.
EDIT:
Also, thank you for writing, I see what you are driving at now.
\(
f \circ g \circ x = f \bullet g \bullet z |_{z=x}\\
\)
Which is the evaluation morphism (?). This makes a lot of sense too. And is definitely a very important distinction. Yes I totally can see the argument for using \( \circ \) here. Correct me if I'm wrong, but can we think of this as,
\(
\text{eval}_{z=x} f \bullet g \bullet z = f\circ g \circ x\\
\)
Where this takes the space of analytic functions, to the space of points. Or something of that nature? Really interesting though. I've never thought of writing it like that; but that's a great distinction notationally.
\(
Y_{\mathcal{P}}(z) = \Omega_{j=0}^{n-1} z + \phi(s_j^*,z)(s_j - s_{j+1})\bullet z\\
\)
Where we assume \( \phi(s,z) \) is some nice function. So as we limit the partition (as it gets finer and finer)--namely \( s_{j+1} - s_j \to 0 \)--under well enough conditions (holomorphy suffices) we get,
\(
\lim_{||\mathcal{P}||\to 0} Y_{\mathcal{P}}(z) = \int_{a}^b \phi(s,z)\,ds\bullet z\\
\)
Now, what this notation means, because we get two things for the price of one, is if,
\(
y(x) = \int_a^x \phi(s,z)\,ds\bullet z\\
\)
Then,
\(
y(a) = z\,\,\,\text{and}\,\,\, y'(x) = \phi(x,y(x))\\
\)
Which is just the equation of a first order differential equation. These motions date allllllllllllll the way back to Euler (usually the bastardization we see is Euler's method, but it's far more advanced). And it's actually kind of funny how much compositional analysis Euler used, which seems to have been just,, well, forgotten... This also helps us learn where the singularities arise--they are where the differential equation blows up.
For instance, take \( \phi(s,z) = z^2 \) then,
\(
\int_a^b z^2 \,ds\bullet z = \frac{1}{\frac{1}{z} + a -b}\\
\)
So where ever \( a-b = \frac{1}{z} \) we know instantly that this infinite composition can't converge because the differential equation blows up. This is really the best I could do--but I did prove some general normality conditions on,
\(
\Omega_{j=0}^{n-1} h_{jn}(s,z)\,\bullet z\\
\)
As we let \( n\to\infty \) then \( h_{jn}(s,z) \) needs to behave either discretely (the infinite composition way) or continuously (which looks something like the riemann-stieljtes composition, but not necessarily).
EDIT:
As to doing this with monoids. I can't even imagine. Maybe something like Tate's thesis with adeles and nonsense. You can count me out for that one, lmao.
EDIT:
Also, thank you for writing, I see what you are driving at now.
\(
f \circ g \circ x = f \bullet g \bullet z |_{z=x}\\
\)
Which is the evaluation morphism (?). This makes a lot of sense too. And is definitely a very important distinction. Yes I totally can see the argument for using \( \circ \) here. Correct me if I'm wrong, but can we think of this as,
\(
\text{eval}_{z=x} f \bullet g \bullet z = f\circ g \circ x\\
\)
Where this takes the space of analytic functions, to the space of points. Or something of that nature? Really interesting though. I've never thought of writing it like that; but that's a great distinction notationally.