Binars

Abbreviation: Bin

Definition

A 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 groupoids. However the more common usage of this term now refers to a category in which each morphism is an isomorphism.

Morphisms

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:

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

Examples

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.

Properties

Classtype variety decidable undecidable no unbounded no no no no no no no no yes yes yes

Finite members

n # of algebras
1 1
2 10
3 3330
4 178981952
5 2483527537094825
6 14325590003318891522275680
7 50976900301814584087291487087214170039
8 155682086691137947272042502251643461917498835481022016

Michael A. Harrison, The number of isomorphism types of finite algebras, Proc. Amer. Math. Soc., 17 1966, 731–737 MRreview