## Generalized effect algebras

Abbreviation: GEAlg

### Definition

A generalized effect algebra is a separation algebra that is

positive: $x\cdot y=e$ implies $x=e=y$.

### Definition

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

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

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

$0$ is an identity: $x+0=x$

$+$ is cancellative: $x+y=x+z$ implies $y=z$ and

$+$ is positive: $x+y=0$ implies $x=0$.

##### Morphisms

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:

### Properties

Classtype first-order

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &2\\ f(4)= &5\\ f(5)= &12\\ f(6)= &35\\ f(7)= &119\\ f(8)= &496\\ f(9)= &2699\\ f(10)= &21888\\ f(11)= &292496\\ \end{array}$