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partial_semigroups [2016/11/26 17:18]
jipsen created
partial_semigroups [2018/08/04 17:55] (current)
jipsen
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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}: $\exists D\subseteq A\times A(\cdot:D\to A)$ and+$\cdot$ is a \emph{partial binary operation}, i.e., $\cdot: A\times A\to A+\{*\}$ and
-$\cdot$ is \emph{associative}: $(x\cdot y)\cdot z\in A$ implies $(x\cdot y)\cdot z=x\cdot (y\cdot z)$ and+$\cdot$ is \emph{associative}: $(x\cdot y)\cdot z\ne *$ or $x\cdot (y\cdot z)\ne *$ imply $(x\cdot y)\cdot z=x\cdot (y\cdot z)$.
-$x\cdot (y\cdot z)\in A$ implies $(x\cdot y)\cdot z=x\cdot (y\cdot z)$.
-
-Remark: $x\cdot y\in A\iff \langle x,y\rangle\in D$
==Morphisms== ==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be partial groupoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: Let $\mathbf{A}$ and $\mathbf{B}$ be partial groupoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
-if $x\cdot y\in A$ then $h(x \cdot y)=h(x) \cdot h(y)$+if $x\cdot y\ne *$ then $h(x \cdot y)=h(x) \cdot h(y)$
====Examples==== ====Examples====
-Example 1: +Example 1: The morphisms is a small category under composition.
====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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====Finite members==== ====Finite members====
+
+http://mathv.chapman.edu/~jipsen/uajs/PSgrp.html
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