# Differences

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categories [2016/11/26 22:28]
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:  $x\circ\text{cod}(x)=x$
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-$\text{dom}(\text{dom}(x))=\text{dom}(x)=\text{cod}(\text{dom}(x))$
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-$\text{cod}(\text{cod}(x))=\text{cod}(x)=\text{dom}(\text{cod}(x))$
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====