conjugative_binars - MathStructures

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=====Conjugative binars=====

Abbreviation: **ConBin**
====Definition====
A \emph{conjugative binar} is a [[binar]] $\mathbf{A}=\langle A,\cdot\rangle$ such that

$\cdot$ is conjugative: $\exists w, \ x\cdot w=y \iff \exists w, \ w\cdot x=y$.

==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: 

====Basic results====


====Properties====
^[[Classtype]]  |  first-order |
^[[Equational theory]]  |   |
^[[Quasiequational theory]]  |   |
^[[First-order theory]]  |   |
^[[Locally finite]]  |  no |
^[[Residual size]]  |   |
^[[Congruence distributive]]  |  no |
^[[Congruence modular]]  |  no |
^[[Congruence n-permutable]]  |  no |
^[[Congruence regular]]  |  no |
^[[Congruence uniform]]  |  no |
^[[Congruence extension property]]  |   |
^[[Definable principal congruences]]  |   |
^[[Equationally def. pr. cong.]]  |   |
^[[Amalgamation property]]  |   |
^[[Strong amalgamation property]]  |   |
^[[Epimorphisms are surjective]]  |   |

====Finite members====

^n  ^  # of algebras^
|1  |  1|
|2  |  4|
|3  |  215|

====Subclasses====
[[Commutative binars]] 

[[Conjugative semigroups]] 

====Superclasses====
[[Binars]]

====References====

Trace: • moufang_quasigroups • quasitrivial_groupoids • ore_domains • complete_distributive_lattices • conjugative_binars • de_morgan_monoids • order_algebras • double_stone_algebras • nonassociative_relation_algebras • left_cancellative_semigroups