separation_algebras - MathStructures

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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====

Trace: • neardistributive_lattices • commutative_ordered_monoids • directed_complete_partial_orders • regular_semigroups • integral_residuated_lattices • partially_ordered_semigroups • finite_commutative_ordered_semigroups • associative_algebras • partial_semigroups • pseudo_mv-algebras