Separation algebras

Abbreviation: SepAlg

Definition

A separation algebra is a generalized separation algebra such that

$\cdot$ is commutative: $x\cdot y = y\cdot x$.

I.e., a separation algebra is a cancellative commutative partial monoid.

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

$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &3\\ f(4)= &8\\ f(5)= &13\\ f(6)= &39\\ f(7)= &120\\ f(8)= &507\\ f(9)= &\\ f(10)= &\\ \end{array}$