Here are some details that are in the spirit of Tarski's work:
Let $M$ be a system of magnitudes and select any element in the carrier set and call it $1$, so that the set $M$ is a pointed set and the object of study becomes $(M,1,+)$. We also have an injective morphism
$\tag 1 \iota: \mathbb N^> = \mathbb N \setminus \{0\} \to M \text{ such that } 1 \mapsto 1$
so we can view the image of the imbedding as an inclusion, $\mathbb N^> \subset M$.
It is not difficult to show that for any $x \in M$ there exist a unique element $H(x)$ such that $H(x)+H(x)=x$.
For any $x \in M$ there exist an $N_x \ge 0$ such that for all $n \ge N_x$ the equations $m H^n(1) + u = x$ with $m \gt 0$ have solutions ('$m \; \text{times}$' is shorthand for repeated addition), so we can take the maximum $m_{(x,n)}$ and define sets $\{m_{(x,n)}H^{n}(1)\} $.
For $s, t \in M$, we can create a set
$\tag 2 X_{(s,t)} = \{\; (m_{(s,n)} \times m_{(t,n)})\, H^{(n+n)}(1) \;\}$
and set $Y_{(s,t)} = \{ m \in M \; | \; (\forall x \in X_{(s,t)}) (\exists u \in M) \,[x + u = m]\}$.
Invoking $\text{P-5}$ (see above link) we can get a $z_{(s,t)} \in M$ that separates $X_{(s,t)}$ and $Y_{(s,t)}$. This element is clearly in $Y_{(s,t)}$ and is therefore unique.
We state the following two theorems without proof.
Theorem 1: The mapping $(s,t) \mapsto z_{(s,t)}$ is a commutative operation that distributes over addition in $(M,1,+)$. Moreover, $1$ is a multiplicative identity in $(M,1,+,*)$.
Theorem 2: Every element $x \in (M,1,+,*)$ has a multiplicative inverse.
