# Differences

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categories [2010/09/04 16:09]
jipsen
categories [2016/11/27 09:54] (current)
jipsen
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$\langle C,\circ\rangle$ is a (large) [[partial semigroup]] $\langle C,\circ\rangle$ is a (large) [[partial semigroup]]
+
+dom amd cod are total unary operations on $C$ such that
$\text{dom}(x)$ is a left unit:  $\text{dom}(x)\circ x=x$ $\text{dom}(x)$ is a left unit:  $\text{dom}(x)\circ x=x$
$\text{cod}(x)$ is a right unit:  $x\circ\text{cod}(x)=x$ $\text{cod}(x)$ is a right unit:  $x\circ\text{cod}(x)=x$
-
-$\text{dom}(\text{dom}(x))=\text{dom}(x)=\text{cod}(\text{dom}(x))$
-
-$\text{cod}(\text{cod}(x))=\text{cod}(x)=\text{dom}(\text{cod}(x))$
if $x\circ y$ exists then $\text{dom}(x\circ y)=\text{dom}(x)$ and $\text{cod}(x\circ y)=\text{cod}(y)$ if $x\circ y$ exists then $\text{dom}(x\circ y)=\text{dom}(x)$ and $\text{cod}(x\circ y)=\text{cod}(y)$
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====Basic results==== ====Basic results====
+$\text{dom}(\text{dom}(x))=\text{dom}(x)=\text{cod}(\text{dom}(x))$
+
+$\text{cod}(\text{cod}(x))=\text{cod}(x)=\text{dom}(\text{cod}(x))$
====Properties==== ====Properties====
-Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.
^[[Classtype]]                        |many-sorted variety  | ^[[Classtype]]                        |many-sorted variety  |
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$\begin{array}{lr}$\begin{array}{lr}
f(1)= &1\\   f(1)= &1\\
-  f(2)= &\\ +  f(2)= &3\\
-  f(3)= &\\ +  f(3)= &11\\
-  f(4)= &\\ +  f(4)= &55\\
-  f(5)= &\\ +  f(5)= &329\\
-\end{array}$+ f(6)= &2858\\ -$\begin{array}{lr} +
-  f(6)= &\\+
f(7)= &\\   f(7)= &\\
f(8)= &\\   f(8)= &\\
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\end{array}$\end{array}$
+http://oeis.org/A125696
====Subclasses==== ====Subclasses====
-  [[...]] subvariety+[[Schroeder categories]]
+
+[[Closed categories]]
-  [[...]] expansion+[[Compact categories]]
====Superclasses==== ====Superclasses====
-  [[...]] supervariety+[[Partially ordered categories]]
-  [[...]] subreduct+[[Partial semigroups]]