−Table of Contents
Binars
Abbreviation: Bin
Definition
A \emph{binar} is a structure A=⟨A,⋅⟩ where ⋅ 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.
Morphisms
Let A and B be binars. A morphism from A to B is a function h:A→B that is a homomorphism:
h(x⋅y)=h(x)⋅h(y)
Examples
Example 1: ⟨N,∧⟩ is the exponentiation binar of the natural numbers, where 0∧0=1. It is not associative nor commutative, and does not have a (two-sided) identity.
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 | 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
