Abbreviation: CBin

A \emph{commutative binar} is a binar $\mathbf{A}=\langle A,\cdot\rangle$ such that

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

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:

$h(x\cdot y)=h(x)\cdot h(y)$

Example 1: $\langle\mathbb N,|\cdot|\rangle$ is the distance binar of the natural numbers, where the binary operation is $|x-y|$.

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

Commutative idempotent binars

Commutative semigroups

Binars