Differences

This shows you the differences between two versions of the page.

commutative_groupoids [2010/07/29 15:46] (current)
Line 1: Line 1:
 +=====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>
 +)]
 +
 +
 +
 +