left_neofield - MathStructures

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=====Left neofileds=====

Abbreviation: **LNfld**
====Definition====
A \emph{left neofield} is a structure $\mathbf{F}=\langle F,+,\backslash,/,0,\cdot,1,^{-1}\rangle $ of type $\langle 2,2,2,0,2,0,1\rangle $ such that


$\langle F,+,\backslash,/,0\rangle $ is a [[loop]]


$\langle F-\{0\},\cdot,1,^{-1}\rangle$ is a [[group]]


$\cdot$ left-distributes over $+$:  $x\cdot(y+z)=x\cdot y+x\cdot z$

==Morphisms==
Let $\mathbf{F}$ and $\mathbf{K}$ be left neofields. A morphism from $\mathbf{F}$
to $\mathbf{K}$ is a function $h:F\to K$ that is a homomorphism: 

$h(x+y)=h(x)+h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(0)=0$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$

====Examples====
Example 1: 

====Basic results====

====Properties====
^[[Classtype]]  | |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Locally finite]]  | |
^[[Residual size]]  | |
^[[Congruence distributive]]  | |
^[[Congruence modular]]  | |
^[[Congruence n-permutable]]  | |
^[[Congruence regular]]  | |
^[[Congruence uniform]]  | |
^[[Congruence extension property]]  | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &\\
f(3)= &\\
f(4)= &\\
f(5)= &\\
f(6)= &\\
\end{array}$

====Subclasses====
[[Neofields]] 

====Superclasses====
[[Loops]] 

[[Groups]]

====References====

Keedwell, A.D., Construction, properties and applications of finite neofields, Comment. Math. Univ. Carolin. 41, 2 (2000) 283–297

[(Ln19xx>
)]

Trace: • temporal_algebras • lie_algebras • dynamic_algebras • complete_lattices • bilattices • steiner_quasigroups • finite_commutative_binars_with_identity_and_zero • generalized_pseudo-effect_algebras • relation_algebras • commutative_lattice-ordered_rings