Table of Contents
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
