commutative_groupoids - MathStructures

This page is read only. You can view the source, but not change it. Ask your administrator if you think this is wrong.

=====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>
)]

Trace: • join-semidistributive_lattices • commutative_bck-algebras • sober_t0-spaces • finite_commutative_integral_ordered_monoids • kleene_logic_algebras • closure_algebras • clifford_semigroups • bands • equations • idempotent_semirings_with_identity_and_zero