# Has this theorem on cancellative monoid actions been discovered and published?

Does a statement equivalent to Theorem 3 below appear in the literature? If it does, what is the earliest published reference?

Theorem 1. Let $$W$$ be a non-trivial cancellative invertible-free  monoid, $$A$$ a non-empty set on which $$W$$ acts on the right, and $$\cal{A}$$ the semicategory with object set $$A$$ and morphisms triples $$(a,w,b)\in A\times (W\backslash\{1\})\times A$$ such that $$b=a\cdot w$$ with composition of morphisms $$(a,w,b)\circ(b,v,c)=(a,wv,c)$$ . Then every morphism of $$\cal{A}$$ is a bimorphism and no morphism is invertible.

Proof. Left as exercise.

Theorem 2. Let $$W$$ and $$A$$ be as above. The quotient of the free monoid in the morphisms of $$\cal{A}$$ by the congruence generated by the composition of morphisms, i.e. $$(a,w,b)(b,v,c)\sim(a,wv,c)$$, is cancellative and invertible-free.

Proof. Left as an exercise.

Denote the monoid in Theorem 2 as $$\cal{A}^\circ$$.

Theorem 3. Let $$W$$ and $$A$$ be as above. Then there exists a cancellative invertible-free monoid $$X$$, set $$B$$ of homographies from $$W$$ to $$X$$, and bijection $$\varphi\colon A\leftrightarrow B$$ that satisfies $$\varphi(a\cdot w)=\varphi(a)^w$$ for all $$a\in A$$ and $$w\in W$$.

Proof. Let $$\cal{A}$$ be as above and let $$X=\cal{A}^\circ$$. Let $$\varphi_a(w)=[a,w,a\cdot w]$$, the equivalence class that contains $$(a,w,a\cdot w)$$, if $$w\not=1$$ and $$\varphi_a(1)=1$$ for all $$a$$.

Note that if $$W=\{1\}$$, then $$\varphi_a=\varphi_b$$ for all $$a,b\in A$$.

 For definitions of these terms, see Can every cancellative invertible-free monoid be embedded in a group? and Is every invertible-free cancellative monoid action represented by "shifting" certain maps?

• You probably want W nontrivial. I suspect that it is known the universal monoid of a cancelative senicategory is cancellative. I dont know what a bimorphism is. I think you are the first person looking at homographies. This seems a nice generalization of what I did for free monoids Jan 3, 2021 at 15:39
• Hello, and good morning. Yes, the construction is similar if not exactly the same. Bimorphisms are defined here  as morphisms that satisfy left and right cancellation. $W$ is assumed non-trivial throughout except for the last sentence, which was intended to be subjunctive. Taking away the identity-preserving requirement, a homography is just a prefix non-decreasing  map.  en.wikipedia.org/wiki/Category_(mathematics)  See p. 501 of Nivat, M., Wirsing, M. (1996). Algebraic Methodology and Software Technology: 5th International Conference, AMAST '96 Munich, Germany, July 1996. Jan 3, 2021 at 16:01