Mathematical Structures: Ordered fields

# Ordered fields

Difference (from prior major revision) (no other diffs)

Changed: 28,30c28,29
 \Large Name of class \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Template}{edit} % Note: replace "Template" with Name_of_class in previous line

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 \abbreviation{Abbr}
 \abbreviation{OFld}

Changed: 35,36c34,36
 A ... is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that
 An ordered field is a structure $\mathbf{F}=\langle F,+,-,0,\cdot,1,\le\rangle$ such that $\langle F,+,-,0,\cdot,1\rangle$ is a \href{Fields.pdf}{field}

Changed: 38c38
 $\langle A,...\rangle$ is a \href{Name_of_class.pdf}{name of class}
 $\langle F,\le\rangle$ is a \href{Linear_orders.pdf}{linear order}

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 $op_1$ is (name of property): $axiom_1$
 $+$ is order-preserving: $x\le y\implies x+z\le y+z$

Changed: 42c42
 $op_2$ is ...: $...$
 $\cdot$ is order-preserving for positive elements: $x\le y\text{ and }0\le z\implies xz\le yz$

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 Classtype & (value, see description) \cite{Ln19xx} \\\hline
 Classtype & universal \\\hline

Changed: 99,115c99
 \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$\qquad $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$
 \begin{finite_members} None

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 \href{....pdf}{...} subvariety \href{....pdf}{...} expansion
 \href{Complete_ordered_fields.pdf}{Complete ordered fields}

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 \href{....pdf}{...} supervariety
 \href{Ordered_rings.pdf}{Ordered rings} reduced type

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 \href{....pdf}{...} subreduct
 \href{Fields.pdf}{Fields} reduced type

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 \bibitem{Ln19xx}
 \bibitem{Lastname19xx}

http://mathcs.chapman.edu/structuresold/files/Ordered_fields.pdf
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\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
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\hyperbaseurl{http://math.chapman.edu/structures/files/}
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\begin{document}
\textbf{\Large Ordered fields}

\abbreviation{OFld}

\begin{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 \href{Fields.pdf}{field}

$\langle F,\le\rangle$ is a \href{Linear_orders.pdf}{linear order}

$+$ is \emph{order-preserving}: $x\le y\implies x+z\le y+z$

$\cdot$ is \emph{order-preserving} for positive elements: $x\le y\text{ and }0\le z\implies xz\le yz$

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.
\end{definition}

\begin{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:
$h(x ... y)=h(x) ... h(y)$
\end{morphisms}

\begin{definition}
An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is ...:  $axiom$

$...$ is ...:  $axiom$
\end{definition}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

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.

\begin{tabular}{|ll|}\hline
Classtype                       & universal \\\hline
Equational theory               & \\\hline
Quasiequational theory          & \\\hline
First-order theory              & \\\hline
Locally finite                  & \\\hline
Residual size                   & \\\hline
Congruence distributive         & \\\hline
Congruence modular              & \\\hline
Congruence $n$-permutable       & \\\hline
Congruence regular              & \\\hline
Congruence uniform              & \\\hline
Congruence extension property   & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong.     & \\\hline
Amalgamation property           & \\\hline
Strong amalgamation property    & \\\hline
Epimorphisms are surjective     & \\\hline
\end{tabular}
\end{properties}
\end{table}

\begin{finite_members} None
\end{finite_members}

\begin{subclasses}\

\href{Complete_ordered_fields.pdf}{Complete ordered fields}

\end{subclasses}

\begin{superclasses}\

\href{Ordered_rings.pdf}{Ordered rings} reduced type

\href{Fields.pdf}{Fields} reduced type

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Lastname19xx}
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 \href{http://www.ams.org/mathscinet-getitem?mr=12a:08034}{MRreview}

\end{thebibliography}

\end{document}
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