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ordered_fields [2010/07/29 15:46] (current)
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+=====Ordered fields=====
+
+Abbreviation: **OFld**
+
+====Definition====
+An \emph{ordered field} is a structure $\mathbf{F}=\langle F,+,-,0,\cdot,1,\le\rangle$ such that
+
+$\langle F,+,-,0,\cdot,1\rangle$ is a [[field]]
+
+$\langle F,\le\rangle$ is a [[linear order]]
+
+$+$ is \emph{order-preserving}: $x\le y\Longrightarrow x+z\le y+z$
+
+$\cdot$ is \emph{order-preserving} for positive elements: $x\le y\text{ and }0\le z\Longrightarrow xz\le yz$
+
+Remark: This is a template.
+
+It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
+
+==Morphisms==
+Let $\mathbf{A}$ and $\mathbf{B}$ be ... . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
+$h(x ... y)=h(x) ... h(y)$
+
+====Definition====
+An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle +...\rangle$ such that
+
+$...$ is ...:  $axiom$
+
+$...$ is ...:  $axiom$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+
+====Properties====
+Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.
+
+^[[Classtype]]                        |universal  |
+^[[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====
+
+====Subclasses====
+  [[Complete ordered fields]]
+
+
+====Superclasses====
+  [[Ordered rings]] reduced type
+
+  [[Fields]] reduced type
+
+
+====References====
+
+[(Lastname19xx>
+F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]]
+)]
+
+

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