Poweroids

[andydude]

The more I think about it, I think it is possible to show that:

• \( ((1, \infty), \mathbb{R}^{+}, x^{x^{y-1}}, 1) \) is a poweroid if the inverse functions are understood a bit more.
  
• \( ((1, \infty), \mathbb{R}^{+}, {\uparrow}{\uparrow}, 1) \) is a poweroid, regardless of the extension. The only requirement I can think of is monotonicity.
  
• \( ((1, \infty), \mathbb{R}^{+}, {\uparrow}{\uparrow}{\uparrow}, 1) \) is a poweroid, etc.
  

I think there are also poweroids with these operations (lower tetration, standard tetration, etc.) over complex domains, such as the one for exponentiation with the unit circle... but I'm not sure how exactly it would work for an arbitrary tetration extension.