binars - MathStructures

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=====Binars=====

Abbreviation: **Bin**
====Definition====
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.

==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.

====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 [[http://www.ams.org/mathscinet-getitem?mr=34 :118|MRreview]]

====Subclasses====
[[Commutative groupoids]] 

[[Idempotent groupoids]] 

[[Semigroups]] 

[[Left-distributive groupoids]] 

====Superclasses====

====References====

[(Ln19xx>
)]

Trace: • relativized_relation_algebras • cancellative_commutative_monoids • multiplicative_lattices • first-order_classes • lattices • finite_semilattices_with_identity • finite_ordered_semigroups • commutative_integral_ordered_monoids • euclidean_domains • equivalence_relations