=====Generalized effect algebras=====

Abbreviation: **GEAlg**

====Definition====
A \emph{generalized effect algebra} is a [[separation algebra]] that is

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

====Definition====
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 $x=0$.

==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 $h(x + y)=h(x) + h(y)$.

====Examples====
Example 1: 

====Basic results====


====Properties====

^[[Classtype]]                        |first-order  |
^[[Equational theory]]                | |
^[[Quasiequational theory]]           | |
^[[First-order theory]]               | |
^[[Locally finite]]                   | |
^[[Residual size]]                    | |
^[[Congruence distributive]]          | |
^[[Congruence modular]]               | |
^[[Congruence $n$-permutable]]        | |
^[[Congruence regular]]               | |
^[[Congruence uniform]]               | |
^[[Congruence extension property]]    | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]      | |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====Finite members====

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

====Subclasses====
[[Effect algebras]]

[[Generalized orthoalgebras]]

====Superclasses====
[[separation algebras]]

[[Generalized pseudo-effect algebras]]

====References====