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groupoids [2010/08/20 10:25]
jipsen
groupoids [2016/11/28 19:02] (current)
jipsen
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=====Groupoids===== =====Groupoids=====
-Abbreviation: **BinOp**+Abbreviation: **Grpd**
====Definition==== ====Definition====
-A \emph{groupoid} is a structure $\mathbf{A}=\langle A,\cdot\rangle$ where +A \emph{groupoid} is a [[category]] $\mathbf{C}=\langle C,\circ,\text{dom},\text{cod}\rangle$ such that
-$\cdot$ is any binary operation on $A$.+
+every morphism is an isomorphism: $\forall x\exists y\ x\circ y=\text{dom}(x)\text{ and }y\circ x=\text{cod}(x)$
==Morphisms== ==Morphisms==
-Let $\mathbf{A}$ and $\mathbf{B}$ be groupoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:  +Let $\mathbf{C}$ and $\mathbf{D}$ be Schroeder categories. A morphism from $\mathbf{C}$ to $\mathbf{D}$ is a function $h:C\rightarrow D$ that is a \emph{functor}: $h(x\circ y)=h(x)\circ h(y)$, $h(\text{dom}(x))=\text{dom}(h(x))$ and $h(\text{cod}(x))=\text{cod}(h(x))$.
-   +
-$h(x\cdot y)=h(x)\cdot h(y)$+Remark: These categories are also called \emph{Brandt groupoids}.
====Examples==== ====Examples====
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====Properties==== ====Properties====
-^[[Classtype]] | variety +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.
-^[[Equational theory]] | decidable +
-^[[Quasiequational theory]] |  +^[[Classtype]]                       |first-order class
-^[[First-order theory]] | undecidable +^[[Equational theory]]               | |
-^[[Locally finite]] | no +^[[Quasiequational theory]]          | |
-^[[Residual size]] | unbounded +^[[First-order theory]]              | |
-^[[Congruence distributive]] | no +^[[Locally finite]]                  | |
-^[[Congruence modular]] | no +^[[Residual size]]                   | |
-^[[Congruence n-permutable]] | no +^[[Congruence distributive]]         | |
-^[[Congruence regular]] | no +^[[Congruence modular]]              | |
-^[[Congruence uniform]] | no +^[[Congruence $n$-permutable]]       | |
-^[[Congruence extension property]] | no +^[[Congruence regular]]              | |
-^[[Definable principal congruences]]  | no +^[[Congruence uniform]]              | |
-^[[Equationally def. pr. cong.]] | no +^[[Congruence extension property]]   | |
-^[[Amalgamation property]] | yes +^[[Definable principal congruences]]  | |
-^[[Strong amalgamation property]] | yes +^[[Equationally def. pr. cong.]]     | |
-^[[Epimorphisms are surjective]] | yes |+^[[Amalgamation property]]           | |
+^[[Strong amalgamation property]]    | |
+^[[Epimorphisms are surjective]]     | |
====Finite members==== ====Finite members====
-^n ^  # of algebras^ +$\begin{array}{lr} -|1 | 1| + f(1)= &1\\ -|2 | 10| + f(2)= &2\\ -|3 | 3330| + f(3)= &3\\ -|4 | 178981952| + f(4)= &7\\ -|5 | 2483527537094825| + f(5)= &9\\ -|6 | 14325590003318891522275680| + f(6)= &16\\ -|7 | 50976900301814584087291487087214170039| + f(7)= &22\\ -|8 | 155682086691137947272042502251643461917498835481022016|+ f(8)= &42\\ + f(9)= &57\\ + f(10)= &90\\ +\end{array}$
-Michael A. Harrison, \emph{The number of isomorphism types of finite algebras}, +http://oeis.org/A140189
-Proc. Amer. Math. Soc., \textbf{17} 1966, 731--737 [[http://www.ams.org/mathscinet-getitem?mr=34 :118|MRreview]]+
====Subclasses==== ====Subclasses====
-[[Commutative groupoids]]  +[[Groups]]
- +
-[[Idempotent groupoids]]  +
- +
-[[Semigroups]]  +
- +
-[[Left-distributive groupoids]] +
====Superclasses==== ====Superclasses====
+[[Categories]]
====References==== ====References====
[(Ln19xx> [(Ln19xx>
+F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]]
)] )]
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