Differences

This shows you the differences between two versions of the page.

generalized_effect_algebras [2018/08/15 10:14] (current)
jipsen created
Line 1: Line 1:
+=====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====
+
+

Toolbox 