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near-rings_with_identity [2010/07/29 15:46] (current)
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 +=====Near-rings with identity=====
 +Abbreviation: **NRng$_1$**
 +====Definition====
 +A \emph{near-ring with identity} is a structure $\mathbf{N}=\langle N,+,-,0,\cdot,1
 +\rangle $ of type $\langle 2,1,0,2,0\rangle $ such that
 +
 +
 +$\langle N,+,-,0,\cdot\rangle $ is a [[near-rings]]
 +
 +
 +$1$ is a \emph{multiplicative identity}:  $x\cdot 1=x\mbox{and}1\cdot x=x$
 +
 +==Morphisms==
 +Let $\mathbf{M}$ and $\mathbf{N}$ be near-rings with identity. A morphism from $\mathbf{M}$
 +to $\mathbf{N}$ is a function $h:M\rightarrow N$ that is a homomorphism:
 +
 +$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$
 +
 +Remark:
 +It follows that $h(0)=0$ and $h(-x)=-h(x)$.
 +
 +====Examples====
 +Example 1: $\langle\mathbb{R}^{\mathbb{R}},+,-,0,\cdot,1\rangle$, the near-ring of functions on the real numbers with pointwise addition, subtraction, zero, composition, and the identity function.
 +
 +
 +====Basic results====
 +$0$ is a zero for $\cdot$: $0\cdot x=0$ and $x\cdot 0=0$.
 +
 +====Properties====
 +^[[Classtype]]  |variety |
 +^[[Equational theory]]  |decidable |
 +^[[Quasiequational theory]]  | |
 +^[[First-order theory]]  | |
 +^[[Locally finite]]  |no |
 +^[[Residual size]]  |unbounded |
 +^[[Congruence distributive]]  |no |
 +^[[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====
 +[[Rings with identity]]
 +
 +====Superclasses====
 +[[Near-rings]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]