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Let's consider the magma $(G,*)$ with infinite elements.

Now I define $\operatorname{left}(G)$ the set of all the left translations

$$\operatorname{left}(G):\{L_a:a \in G ,L_a(b)=a*b\}$$

And $iter(a)$ as the set formed by the left traslation by $a\in G$ and closed under function composition or in other words the iterations of the left translations.

$$\operatorname{iter}(a):\{L_a^{n}:n \in \Bbb N\setminus\{0\}\}$$

What are the weakest conditions that $(G,*)$ must satisfies if we want that exist an collection injective functions $\mathcal F_a:\operatorname{iter}(a)\rightarrow > \operatorname{left}(G)$? $\mathcal F_a$ are defined in this way

$$\mathcal F_a(L_a^1)=L_a$$ $$\mathcal F_a(L_a^n)=L_{a'}$$ $$\mathcal (\mathcal F_a(L_a^n))(x)=L_a^n(x)$$

I think that this is equivalent to the this statement

$$\operatorname{iter}(a)\subseteq \operatorname{left}(G)$$.

If $(G,*)$ is associative then this always holds because $L_a(L_a(x))=L_{a*a}(x)$ and in general $\mathcal F_a(L_a^n)=L_{a^n}$

But this assumption is too stroong for my needs

A weaker condition is that if $*$ is right invertible (exists a $R_a^{-1}$ such that $R_b^{-1}(a*b)=(a*b)\setminus_* b=a$) then we can define the functions $\beta_a$

$$\beta_a(n)*b=a_1*\cdots(a_n*b)$$ $$\beta_a(n)=(a_1*\cdots(a_n*b))\setminus_* b$$

$$\beta_a(n)=R_b(L_a^n(b))$$

and these functions MUST be always constants (for every $b$) so we can define the injections $\mathcal F_a$ :

$$L^n_a=L_{\beta_a(n)}$$

and thus $ \mathcal F_a$ exists

$$\mathcal F_a(L_a^n)=L_{\beta_a(n)}$$

$1$-$\beta_a$ not depend from $b$ and is always costant, $\mathcal F_a$ exists and $\operatorname{iter}(a)\subseteq \operatorname{left}(G)$ are three equivalent statements?

$3$-When $\beta_a$ satisfies that weak condition?

$3$-If $(G,*)$ is not commutative, not associative and not left invertible but is right invertible is possible that this bijection from $\operatorname{iter}(a)\rightarrow \operatorname{left}(g)$ exists? Or maybe there can be a surjection?

Note:

the image of $\operatorname{iter}(a)$ by$\mathcal F_a$, $\mathcal F_a[\operatorname{iter}(a)]$ should be always a commutative subsemigroup of $(\operatorname{left}(G),\circ)$, commutative submonoid if $*$ has a left unit and a commutative subgroup if $*$ is right invertible.

Update: here a related question.

Existence of an operation $\cdot$ such that $(a*(b*c))=(a\cdot b)*c$

Seems to me that the conditiont that the User Goos found is really similar to inclusion condition.

Anyways I think that the questions here are still meaningfull.

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