In Faltin-Metropolis-Ross-Rota's [FMRR] paper *The Real Numbers as a Wreath Product* [Adv. Math. 16(3), 278-304 (1975)], the real numbers are constructed as a quotient of a certain subset of the ring of formal Laurent series $\mathbb{Z}((T))$, emphasizing the digit-expansion aspect of elements of $\mathbb{R}$ by formalizing the idea of "infinite carrying". The paper by itself is not hard to follow, and my question is actually more about **the title** of the paper, for even after reading it completely, I did not understand what exactly was being referred to as a wreath product. (The term *wreath product* is mentioned exactly once in the entire paper: in the title!)

Here's a (very) quick summary of the construction:

Let $b\geq 2$ be a natural number (

base). Write $\mathbf{C}_b \subseteq \mathbb{Z}((T))$ for thering of convergent sequences:$$ \mathbf{C}_b = \left\{ \sum_{n\in\mathbb{Z}} a_n T^n \in \mathbb{Z}((T)) ~\bigg|~ |a_n| = o(b^n) \right\}. $$ Write $\mathfrak{K}_b := 1 - bT$ for thecarry constant. Then, denote thering of bounded sequencesand thecarry idealby, resp., $$ R_b := \left\{ \sum_{j\in\mathbb{Z}} a_j T^j \in \mathbb{Z}((T)) ~\bigg|~ \sum_{j=1}^{n} |a_j| b^{n-j} = O(b^n) \right\}, \quad I_b := (\mathfrak{K}_b \mathbf{C}_b) \cap R_b, $$ so that $I_b \subseteq R_b \subseteq \mathbf{C}_b$. Then:

Theorem(FMRR).$R_b/I_b \simeq \mathbb{R}$ as ordered topological fields, for every $b \geq 2$.

Remark.The big-$O$ notation is just a shorthand for the appropriate condition (which, although similar, is defined solely in terms of $\mathbb{Z}$). Both the order and the topology turn out to be the "obvious ones" (of course); while a topology may be defined in terms of the magnitude of $\sum_j |\alpha_j - \beta_j| b^{n-j}$ as $n$ grows, the order depends on a canonical representation of strings (referred to asclearingin [FMRR]).

My question is: what exactly in this construction is being referred to as a **wreath product**? I see that $\prod_{j\in\mathbb{Z}} \mathbb{Z}/(b)$ could be acting as a basis for some potential wreath product, but I see no obvious group action that makes it substantially different from the generalized Lamplighter group $\mathbb{Z}/(b) \,\wr\, \mathbb{Z}$.