=====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>
)]