Abbreviation: GSepAlg

A \emph{generalized separation algebra} is a cancellative partial monoid such that

$\cdot$ is \emph{conjugative}: $\exists w, \ x\cdot w=y \iff \exists w, \ w\cdot x=y$.

Let $\mathbf{A}$ and $\mathbf{B}$ be cancellative partial monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(e)=e$ and if $x\cdot y\ne *$ then $h(x \cdot y)=h(x) \cdot h(y)$.

Example 1:

$\begin{array}{lr}

f(1)= &1\\
f(2)= &2\\
f(3)= &3\\
f(4)= &8\\
f(5)= &14\\
f(6)= &48\\
f(7)= &172\\
f(8)= &\\
f(9)= &\\
f(10)= &\\

\end{array}$

Separation algebras

Generalized pseudo-effect algebras

Cancellative partial monoids