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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>
+)]

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