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partial_groupoids [2016/11/26 17:11]
jipsen
partial_groupoids [2018/08/04 18:07] (current)
jipsen
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A \emph{partial groupoid} is a structure $\mathbf{A}=\langle A,\cdot\rangle$, where A \emph{partial groupoid} is a structure $\mathbf{A}=\langle A,\cdot\rangle$, where
-$\cdot$ is a \emph{partial binary operation}: $\exists D\subseteq A\times A(\cdot:D\to A)$.+$\cdot$ is a \emph{partial binary operation}, i.e., $\cdot: A\times A\to A+\{*\}$.
-Remark: $x\cdot y\in A\iff \langle x,y\rangle\in D$ +Remark: The domain of definition of $\cdot$ is Dom$(\cdot)=\{\langle x,y\rangle\in A^2 \mid x\cdot y\ne *\}$
==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====
Line 45: Line 45:
  f(1)= &2\\   f(1)= &2\\
  f(2)= &45\\   f(2)= &45\\
-  f(3)= &\\ +  f(3)= &43968\\ 
-  f(4)= &\\ +  f(4)= &6358196250\\ 
-  f(5)= &\\+  f(5)= &236919104155855296\\
\end{array}$     \end{array}$    
-$\begin{array}{lr} 
-  f(6)= &\\ 
-  f(7)= &\\ 
-  f(8)= &\\ 
-  f(9)= &\\ 
-  f(10)= &\\ 
-\end{array}$ 
 +See http://oeis.org/A090601
====Subclasses==== ====Subclasses====