Abbreviation: GEAlg

A \emph{generalized effect algebra} is a separation algebra that is

\emph{positive}: $x\cdot y=e$ implies $x=e=y$.

A \emph{generalized effect algebra} is of the form $\langle A,+,0\rangle$ where $+:A^2\to A\cup\{*\}$ is a partial operation such that

$+$ is \emph{commutative}: $x+y\ne *$ implies $x+y=y+x$

$+$ is \emph{associative}: $x+y\ne *$ implies $(x+y)+z=x+(y+z)$

$0$ is an \emph{identity}: $x+0=x$

$+$ is \emph{cancellative}: $x+y=x+z$ implies $y=z$ and

$+$ is \emph{positive}: $x+y=0$ implies $x=0$.

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

Example 1:

$\begin{array}{lr}

f(1)= &1\\
f(2)= &1\\
f(3)= &2\\
f(4)= &5\\
f(5)= &12\\
f(6)= &35\\
f(7)= &119\\
f(8)= &496\\
f(9)= &2699\\
f(10)= &21888\\
f(11)= &292496\\

\end{array}$

Effect algebras

Generalized orthoalgebras

separation algebras

Generalized pseudo-effect algebras