# Is every cyclic right action of a cancellative invertible-free monoid on a set isomorphic to the set of shifts of some homography?

The terms are defined in a related question. 

Conjecture 1. Let $$A$$ be a set, $$W$$ a cancellative invertible-free monoid, and $$\cdot\colon A\times W\rightarrow A$$ a cyclic right $$W$$-action generated by an element $$a_0 \in A$$, in symbols $$A=a_0\cdot W=\{a_0\cdot w\colon w\in W\}$$. Then there exists a cancellative invertible-free monoid $$X$$, homography $$b\colon W\rightarrow X$$, and bijection $$\varphi\colon A\leftrightarrow b^W$$ satisfying $$\varphi(a_0)=b$$ and $$\varphi(a\cdot w)=\varphi(a)^w$$ for all $$a\in A$$ and $$w\in W$$.

Conjecture 2. Let $$R$$ be a commutative ring, $$W$$ a cancellative invertible-free monoid, $$F$$ an $$R$$-module, and $$\cdot\colon F\times W\rightarrow F$$ a cyclic $$R$$-linear $$W$$-action linearly generated by $$f_0\in F$$, in symbols $$R\,f_0\cdot W=F$$. Then there exists a cancellative invertible-free monoid $$X$$, homography $$g\colon W\rightarrow X$$, and $$R$$-module isomorphism $$U\colon F\leftrightarrow R[g^W]$$ satisfying $$U(f_0)=g$$ and $$U(f\cdot w)=T_w\,U(f)$$ for all $$f\in F$$ and $$w\in W$$.

• Besides free monoids do you have any examples where this is true? Dec 29, 2020 at 2:06
• Here is one: non-negative reals acting on $A=[-1,1]\subset\mathbb{R}$ by $a\cdot x=a+x-⌊a+x⌋$ if $a+x>0$ and $a\cdot x = a+x$ otherwise. Let $f$ be the function on the non-negative reals equal to the sum of the identity and floor. Then set $g(x) = f(x-1)$ on $[1,\infty)$ and $g(x)=0$ on $[0,1)$ and we have $g^{\mathbb{R_+}}\cong A$ as actions. Dec 29, 2020 at 15:59