semifields - MathStructures

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=====Semifields=====

Abbreviation: **Sfld**
====Definition====
A \emph{semifield} is a [[semirings with identity|semiring with identity]] $\mathbf{S}=\langle S,+,\cdot, 1\rangle $ such that


$\langle S^*,\cdot,1\rangle$ is a group, where $S^*=S-\{0\}$ if $S$ has an absorbtive $0$, and $S=S^*$ otherwise.


==Morphisms==
Let $\mathbf{S}$ and $\mathbf{T}$ be semifields. A morphism from $\mathbf{S}$
to $\mathbf{T}$ is a function $h:S\to T$ that is a homomorphism: 

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

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

====Basic results====

The only finite semifield that is not a field is the 2-element Boolean semifield: https://arxiv.org/pdf/1709.06923.pdf 

====Properties====
^[[Classtype]]  | |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Locally finite]]  |No |
^[[Residual size]]  |Unbounded |
^[[Congruence distributive]]  |No |
^[[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====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &2\\
f(3)= &1\\
f(4)= &1\\
f(5)= &1\\
f(6)= &0\\
\end{array}$

====Subclasses====
[[Fields]] 

====Superclasses====
[[Semirings with identity]] 


====References====

[(Ln19xx>
)]

Trace: • flec-algebras • abelian_ordered_groups • finite_commutative_semigroups_with_zero

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