I am looking for the names (and/or for any information regarding) two algebras, one "free" and one "restricted" by an equivalence class.
In both cases, there is an (infix) binary product operator "$.$" for which we have two (prefix) unary operators $\alpha$ and $\beta$ such that $\alpha (x . y) = x$ and $\beta (x . y) = y$ for all $x$ and $y$ in some set $S$, and $x . y \in S$. The product is neither associative not commutative, and there is no identity element. If it helps, you may understand those in Lisp-like terms: the product is cons; $\alpha$ is car, head or first; and $\beta$ is cdr tailor rest. Unlike Lisp-like languages, there are no atoms, and cycles are allowed.
The elements of $S$ can also be understood as vertices in a labelled directed acyclic graphs where all vertices have out-degree 1 for both $\alpha$-edges and $\beta$-edges (so total out-degree 2), and there are no restrictions to in-degree. I think of this as "a duo of directed pseudoforests".
$$ a = a . a \\ b = b . c \\ c = c . b \\ d = d . d \\ e = a . b \\ f = d . c \\ g = d . b \\ h = e . g \\ i = e . f \\ j = g . j \\ k = d . k $$ The above shows an example of the "free" case, depicted here because I can't embed images yet. I now introduce an equivalence class such that any distinction between two vertices has to emerge from the vertices reachable from them. In the restricted algebra, one may thus deduce that $a = d$ and $e = g$ (but $e \neq f$). Here is a depiction of the result.
The easiest way I've found to explicit this equivalence class is to obtain a canonical representation (and total ordering) for each Strongly Connected Component (SCC) and for their vertices. Except for the trivial case of an SCC with a single vertex and no loop, it is guaranteed that there is at least one directed Hamiltonian cycle in the SCC. Each such cycle can be associated with an $n$-bead necklaces with 2 colors when turning over is not allowed, where the colors are the edge labels $\alpha$ and $\beta$. My canonical representations are built around those necklaces. I'm skipping details for the sake of brevity, but I can expand if there is interest.
Before I reinvent the wheel any further, do you know how much of this already exists? Under what name(s) is it known? Or am I making some obvious mistake? Thanks!
