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partial_groupoids [2010/07/29 15:46]
127.0.0.1 external edit
partial_groupoids [2018/08/04 18:07] (current)
jipsen
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=====Partial groupoids===== =====Partial groupoids=====
-% Note: replace "Template" with Name_of_class in previous line 
-Abbreviation: **PBinOp**+Abbreviation: **Pargoid**
====Definition==== ====Definition====
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$(\cdot)=\{\langle x,y\rangle\in A^2 \mid x\cdot y\ne *\}$
- +
-This is a template. +
-If you know something about this class, click on the 'Edit text of this page' link at the bottom and fill out this page. +
- +
-It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.+
==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 $h(x \cdot y)=h(x) \cdot h(y)$
- +
-====Definition==== +
-An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle +
-...\rangle$ such that +
- +
-$...$ is ...:  $axiom$ +
-   +
-$...$ is ...:  $axiom$+
====Examples==== ====Examples====
-Example 1: +Example 1: The empty partial binary operation on any set $A$ gives a partial groupoid.
====Basic results==== ====Basic results====
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====Properties==== ====Properties====
-Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described. 
^[[Classtype]]                        |first-order  | ^[[Classtype]]                        |first-order  |
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$\begin{array}{lr} $\begin{array}{lr}
-  f(1)= &1\\ +  f(1)= &2\\ 
-  f(2)= &\\ +  f(2)= &45\\ 
-  f(3)= &\\ +  f(3)= &43968\\ 
-  f(4)= &\\ +  f(4)= &6358196250\\ 
-  f(5)= &\\+  f(5)= &236919104155855296\\
\end{array}$     \end{array}$    
-$\begin{array}{lr} 
-  f(6)= &\\ 
-  f(7)= &\\ 
-  f(8)= &\\ 
-  f(9)= &\\ 
-  f(10)= &\\ 
-\end{array}$ 
 +See http://oeis.org/A090601
====Subclasses==== ====Subclasses====
-  [[Groupoids]]+[[Groupoids]]
-  [[Partial semigroups]]+[[Partial semigroups]]
====Superclasses==== ====Superclasses====
-  [[Ternary relations]]+[[Ternary relations]]
====References==== ====References====
-[(Lastname19xx+[(Ljapin1997
-F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] +E. S. Ljapin and A. E. Evseev, \emph{The theory of partial algebraic operations}, Kluwer, 1997
)] )]