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generalized_effect_algebras [2018/08/15 10:14] (current)
jipsen created
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 +=====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====
 +
 +