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commutative_groupoids [2010/07/29 15:46] (current)
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+=====Commutative Groupoids=====
+
+Abbreviation: **CBinOp**
+====Definition====
+A \emph{commutative groupoid} is a structure $\mathbf{A}=\langle A,\cdot\rangle$ where
+$\cdot$ is any commutative binary operation on $A$, i.e.
+$x\cdot y=y\cdot x$
+
+==Morphisms==
+Let $\mathbf{A}$ and $\mathbf{B}$ be commutative groupoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
+
+$h(x\cdot y)=h(x)\cdot h(y)$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+
+====Properties====
+^[[Classtype]]  |  variety |
+^[[Equational theory]]  |  decidable |
+^[[Quasiequational theory]]  |   |
+^[[First-order theory]]  |  undecidable |
+^[[Locally finite]]  |  no |
+^[[Residual size]]  |  unbounded |
+^[[Congruence distributive]]  |  no |
+^[[Congruence modular]]  |  no |
+^[[Congruence n-permutable]]  |  no |
+^[[Congruence regular]]  |  no |
+^[[Congruence uniform]]  |  no |
+^[[Congruence extension property]]  |  no |
+^[[Definable principal congruences]]  |  no |
+^[[Equationally def. pr. cong.]]  |  no |
+^[[Amalgamation property]]  |  yes |
+^[[Strong amalgamation property]]  |  yes |
+^[[Epimorphisms are surjective]]  |  yes |
+====Finite members====
+
+$\begin{array}{lr} + f(1)= &1\\ + f(2)= &\\ + f(3)= &\\ + f(4)= &\\ + f(5)= &\\ + f(6)= &\\ +\end{array}$
+
+====Subclasses====
+  [[Commutative semigroups]]
+
+  [[Idempotent commutative groupoids]]
+
+  [[Commutative left-distributive groupoids]]
+
+====Superclasses====
+  [[Groupoids]]
+
+
+====References====
+
+[(Ln19xx>
+)]
+
+
+
+

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