# Differences

This shows you the differences between two versions of the page.

near-fields [2010/07/29 15:46] (current)
Line 1: Line 1:
+=====Near-fields=====
+Abbreviation: **NFld**
+====Definition====
+A \emph{near-field} is a [[near-rings with identity]] $\mathbf{N}=\langle N,+,-,0,\cdot,1 +\rangle$ such that
+
+
+$\mathbf{N}$ is non-trivial:  $0\ne 1$
+
+
+every non-zero element has a multiplicative inverse:  $x\ne 0\Longrightarrow \exists y +(x\cdot y=1)$
+
+Remark:
+The inverse of $x$ is unique, and is usually denoted by $x^{-1}$.
+
+
+==Morphisms==
+Let $\mathbf{M}$ and $\mathbf{N}$ be near-fields. A morphism from $\mathbf{M}$
+to $\mathbf{N}$ is a function $h:M\rightarrow N$ that is a homomorphism:
+
+$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$
+
+Remark:
+It follows that $h(0)=0$ and $h(-x)=-h(x)$.
+
+====Examples====
+Example 1:
+
+====Basic results====
+$0$ is a zero for $\cdot$: $0\cdot x=0$ and $x\cdot 0=0$.
+
+====Properties====
+^[[Classtype]]  |first-order |
+^[[Equational theory]]  | |
+^[[Quasiequational theory]]  | |
+^[[First-order theory]]  | |
+^[[Locally finite]]  |no |
+^[[Residual size]]  |unbounded |
+^[[Congruence distributive]]  | |
+^[[Congruence modular]]  |yes |
+^[[Congruence n-permutable]]  |yes, $n=2$ |
+^[[Congruence regular]]  |yes |
+^[[Congruence uniform]]  |yes |
+^[[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====
+[[Fields]]
+
+====Superclasses====
+[[Near-rings with identity]]
+
+
+====References====
+
+[(Ln19xx>
+)]