# Differences

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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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