# Differences

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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. +
- +
-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$gives a partial groupoid. ====Basic results==== ====Basic results==== Line 35: Line 21: ====Properties==== ====Properties==== -Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described. ^[[Classtype]] |first-order | ^[[Classtype]] |first-order | Line 58: Line 43:$\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
)] )]