Table of Contents
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]]
