Commutative binars

Abbreviation: CBin

Definition

A 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

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

Superclasses

References