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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>
 +)]