The terms are defined in a related question. [1]

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 $w\in W$.

[1] Is every invertible-free cancellative monoid action represented by "shifting" certain maps?