Abbreviation: Bin
A \emph{binar} is a structure $\mathbf{A}=\langle A,\cdot\rangle$ where $\cdot$ is any binary operation on $A$.
Remark: In Universal Algebra binars are also called \emph{groupoids}. However the more common usage of this term now refers to a category in which each morphism is an isomorphism.
Let $\mathbf{A}$ and $\mathbf{B}$ be 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: $\langle\mathbb N,{}^\wedge\rangle$ is the exponentiation binar of the natural numbers, where $0{}^\wedge0=1$. It is not associative nor commutative, and does not have a (two-sided) identity.
| 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 |
| n | # of algebras |
|---|---|
| 1 | 1 |
| 2 | 10 |
| 3 | 3330 |
| 4 | 178981952 |
| 5 | 2483527537094825 |
| 6 | 14325590003318891522275680 |
| 7 | 50976900301814584087291487087214170039 |
| 8 | 155682086691137947272042502251643461917498835481022016 |
Michael A. Harrison, \emph{The number of isomorphism types of finite algebras}, Proc. Amer. Math. Soc., \textbf{17} 1966, 731–737 MRreview