# Differences

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separation_algebras [2018/08/04 22:44] (current)
jipsen created
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+=====Separation algebras=====
+
+Abbreviation: **SepAlg**
+
+====Definition====
+A \emph{separation algebra} is a [[generalized separation algebra]] such that
+
+$\cdot$ is \emph{commutative}: $x\cdot y = y\cdot x$.
+
+I.e., a separation algebra is a cancellative commutative partial monoid.
+
+==Morphisms==
+Let $\mathbf{A}$ and $\mathbf{B}$ be cancellative partial monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
+$h(e)=e$ and
+if $x\cdot y\ne *$ then $h(x \cdot y)=h(x) \cdot h(y)$.
+
+====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====
+
+$\begin{array}{lr} + f(1)= &1\\ + f(2)= &2\\ + f(3)= &3\\ + f(4)= &8\\ + f(5)= &13\\ + f(6)= &39\\ + f(7)= &120\\ + f(8)= &507\\ + f(9)= &\\ + f(10)= &\\ +\end{array}$
+
+====Subclasses====
+[[Generalized effect algebras]]
+
+[[Generalized pseudo-effect algebras]]
+
+
+====Superclasses====
+[[Generalized separation algebra]]
+
+
+====References====
+
+

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