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partial_semigroups [2018/07/23 16:57]
jipsen
partial_semigroups [2018/08/04 17:55] (current)
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A \emph{partial semigroup} is a structure $\mathbf{A}=\langle A,\cdot\rangle$, where A \emph{partial semigroup} is a structure $\mathbf{A}=\langle A,\cdot\rangle$, where
-$\cdot$ is a \emph{partial binary operation}, i.e., $\cdot: A\times A\to A+\{*\})$ and+$\cdot$ is a \emph{partial binary operation}, i.e., $\cdot: A\times A\to A+\{*\}$ and
$\cdot$ is \emph{associative}: $(x\cdot y)\cdot z\ne *$ or $x\cdot (y\cdot z)\ne *$ imply $(x\cdot y)\cdot z=x\cdot (y\cdot z)$. $\cdot$ is \emph{associative}: $(x\cdot y)\cdot z\ne *$ or $x\cdot (y\cdot z)\ne *$ imply $(x\cdot y)\cdot z=x\cdot (y\cdot z)$.
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====Basic results==== ====Basic results====
+Partial semigroups can be identified with [[semigroups with zero]] since for any partial semigroup $A$ we can define a semigroup $A_0=A\cup\{0\}$ (assuming $0\notin A$)
+and extend the operation on $A$ to $A_0$ by $0x=0=x0$ for all $x\in A$. Conversely, given a semigroup with zero, say $B$, define a partial semigroup
+$A=B\setminus\{0\}$ and for $x,y\in A$ let $xy=*$ if $xy=0$ in $B$. These two maps are inverses of each other.
+However, the category of partial semigroups is not the same as the category of semigroups with zero since the morphisms differ.
====Properties==== ====Properties====

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