# Differences

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euclidean_domains [2010/07/29 15:46] (current)
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+=====Euclidean Domains=====
+Abbreviation: **EucDom**
+====Definition====
+A \emph{Euclidean domain} is an [[integral domains]] $\langle D,+,-,0,\cdot,1\rangle$ together with a function $d:D\setminus\{0\} \to\mathbf{N}$ such that
+
+
+$\forall a,b\ (a\ne 0$, $b\neq 0 \Longrightarrow d(a)\le d(ab))$
+
+
+$\forall a,b \exists q,r\ (a=b\cdot q+r$, $(r=0 \mbox{or} d(r)<d(b)))$
+
+==Morphisms==
+
+====Examples====
+Example 1: $\langle\mathbb{Z},+,-,0,\cdot,1,d\rangle$, the ring of integers with addition, subtraction, zero, and multiplication is a Euclidean domain with $d(a)=|a|$.
+
+
+====Basic results====
+
+====Properties====
+^[[Classtype]]  |first-order |
+^[[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====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &1\\ +f(3)= &1\\ +f(4)= &1\\ +f(5)= &1\\ +f(6)= &0\\ +\end{array}$
+
+====Subclasses====
+[[Fields]]
+
+====Superclasses====
+[[Principal Ideal Domains]]
+
+
+====References====
+
+[(Ln19xx>
+)]

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