=====Commutative binars=====

Abbreviation: **CBin**
====Definition====
A \emph{commutative binar} is a [[binar]] $\mathbf{A}=\langle A,\cdot\rangle$ such that

$\cdot$ is commutative: $x\cdot y=y\cdot x$.

==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be commutative binars. 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: $\langle\mathbb N,|\cdot|\rangle$ is the distance binar of the natural numbers, where the binary operation is $|x-y|$. 

====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====

^n  ^  # of algebras^
|1  |  1|
|2  |  4|
|3  |  129|
|4  |  43968|
|5  |  254429900|
|6  |  30468670170912|
|7  |  91267244789189735259|
|8  |  8048575431238519331999571800|
|9  |  24051927835861852500932966021650993560|
|10  |  2755731922430783367615449408031031255131879354330|

see [[finite commutative binars]] and http://www.research.att.com/~njas/sequences/A001425

====Subclasses====
[[Commutative idempotent binars]] 

[[Commutative semigroups]] 

====Superclasses====
[[Binars]]

====References====