Generalized separation algebras

Abbreviation: GSepAlg

Definition

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

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

Morphisms

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:

Properties

Classtype first-order

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &3\\ f(4)= &8\\ f(5)= &14\\ f(6)= &48\\ f(7)= &172\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$