Geometry of rings and semi-rings

Question

Basically, I wonder whether a theory similar to geometric group theory has been or could be developed for rings and semirings.

One direction would be the following. Consider $\mathbb{N}$ (with the French convention, i.e. including $0$), and let $n\in\mathbb{N}$. Consider all algorithms based on the following operations:

• adding $1$,
• multiplying two variables,
• summing two variables

and allowed to have any number of local variables (no loops or other advanced programming, just allocation of variables -for free- and the above three operations). Let $\delta(n)$ be the least number of operations needed for such an algorithm to return $n$.

What can we say about the function $\delta$? Can it be generalized to other rings or semi-rings? Can it be turned into a distance? Does it say something about the algebra of the given ring or semi-ring?

(the emphasis was added in edit, this is the part I am most interested in).

It feels a little bit like Kolmogorov complexity (except that I chose not to allow taking differences of variables, which is questionable).

Edit Another direction would be to consider the minimal numbers $||n||$ of $1$s needed to write $n$ by a well-formed formula with the symbols $1$ $+$ $\times$ $($ and $)$; this happens to lead to open questions, see this answer on MO.

There could be many other ways to turn finitely generated rings or semi-rings into geometric object, I would be interested in any of them.

Added in edit To clarify where I'm headed to, both cases show that there are natural ways to construct functions that can play the role of the distance to $0$; but I am not aware of any distance functions (taking two arguments) of this sort. Is there any that has been considered?

Answer 1

(1) As already mentioned in the comments by Emil Jeřábek, your questions about $\mathbb{N}$ are studied under the name of straight-line programs or addition-multiplication chains. They are closely related to major complexity questions, e.g. if you cannot compute $n!$ using only $O((\log n)^c)$ such operations for any constant $c$, then Valiant's conjecture follows ($\mathsf{VP} \neq \mathsf{VNP}$) [Burgisser, comput. complex. 2009]. In a ring you can turn circuit size in the generators into a metric $d(f,g)$ by considering the circuit size of $f-g$.

(2) While it's not a metric, Gelfand-Kirillov dimension of a ring is in the same spirit as geometric group theory, but for rings. It is basically the exponent of the polynomial growth rate (if it is bounded by a polynomial) of balls in the "Cayley graph" of the ring.