%%  Sample article for Algebra Universalis: AUart.tex
%%
%%  May, 2010

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%%  PREAMBLE                                                     %%
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\documentclass{au}
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%% For completion by editorial office:
 \presentedby{\dots}
 \received{\dots}{\dots}
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\usepackage{amsmath,amssymb,url}
%\usepackage{graphics} %%% (Optional. Use for incorporating separate graphics files.)

\numberwithin{equation}{section}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{remark}[theorem]{Remark}

%% Sample macros %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \DeclareMathOperator{\dom}{dom}

 \newcommand{\N}{{\mathbb{N}}}
 \newcommand{\B}{{\mathbf{B}}}
 \newcommand{\inv}[2]{#1^{-1}(#2)}

 \renewcommand{\le}{\leqslant}

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\begin{document}

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%%  FRONT MATTER                                                 %%
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\title[Sample article]{A sample article for Algebra Universalis}

%% First author (Note: The order of the items here is important!)
\author[G. A. Menuhin]{George A. Menuhin}
\email{menuhin@cs.uwinnebago.edu}
\urladdr{http://www.cs.uwinnebago.edu/homepages/menuhin}
\address{Computer Science Department\\
University of Winnebago\\Minnesota 23714\\USA}

%% Second author (Note: The order of the items here is important!)
\author[I. M. Coauthor]{Ivan M. Coauthor}
\email{coauthor@eh.com.ca}
\address{77 Sunset Strip\\Elk Haunches\\Alberta H0H 0H0\\Canada}

%% Thanks (Optional)
\thanks{The research of the first author was supported by the NSF under
grant no.~3.14159. The second author thanks his long-suffering
family.}

%% Dedication (Optional)
\dedicatory{This article is dedicated to AU authors, present and future}

%% AMS subject classification; see http://www.ams.org/msc
%% Only one Primary. Possibly several Secondary.
\subjclass[2010]{Primary: 00A99; Secondary: 08A40, 06E30.}

%% Keywords and phrases
\keywords{partial examples, distributed instructions}


\begin{abstract}
In this note we give instructions for preparing the \LaTeX\
file of an AU paper.
\end{abstract}

\maketitle

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%%  MAIN MATTER                                                  %%
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\section{Introduction}\label{S:intro}

This sample article gives instructions for preparing a \LaTeX\
file in the AU format. It also serves as a template: you should
model your own \LaTeX\ file on this.

The following sections discuss the various parts of the \LaTeX\
file, in turn; except for Section~\ref{S:ex}, which gives
examples of some constructions that you may need throughout
your paper. If you are unfamiliar with \LaTeX, you may also
need to refer to one of the several available
texts~\cite{More,Comp}, or start with George Gr\"atzer's freely
downloadable \emph{Short Course}~\cite{Short} (the pdf file,
the sample \LaTeX\ files, and the video presentations).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preamble}

You will need the class file \texttt{au.cls}, which can be
downloaded from the AU homepage (Section~D of
\emph{Instructions for Authors}):
\begin{center}
http://www.math.umanitoba.ca/homepages/au
\end{center}
You do not have to learn anything new about \texttt{au.cls} to
use it. It is a modification of the standard AMS article class
\texttt{amsart.cls}; the same rules apply.

Place the file \texttt{au.cls} in your article's folder (or in
the input directory of your \TeX\ implementation), and start
your article with \verb+\documentclass{au}+.

Please base your \LaTeX\ file's preamble on that for this
sample article. In particular, note the following AU
conventions.
\begin{itemize}
 \item We use \verb+\theoremstyle{plain}+ for statements
     that are usually proved (Theorem, Lemma, etc.); we use
     \verb+\theoremstyle{definition}+ for statements that
     are usually not proven (Definition, Remark, etc.). We
     do not use \verb+\theoremstyle{remark}+.
 \item Unless a paper is very short, theorems, definitions,
     etc.\ and numbered equations should be numbered by
     section. Theorems, definitions, etc.\ should
     be numbered consecutively, not independently. So
     Theorem~3.6 is followed by Lemma~3.7, for example.
 \item To define your own commands, use the \LaTeX\
     commands \verb+\newcommand+ and \verb+\renewcommand+;
     do not use \verb+\def+.
\end{itemize}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Front matter}\label{S:Fm}

Please base your \LaTeX\ file's front matter on that used for
this article. The front matter consists of the following parts.
\begin{description}
 \item[Title] Only the first letter of the first word and
     the first letters of proper names are capitalized.
 \item[Authors] Each author is listed separately, along
     with e-mail and address. (Note: The order of the items
     is important!)
 \item[Thanks] At most one per author, listed together  so
     that if two authors give thanks, it reads: ``The first
     author thanks blah, blah, blah. The second author
     thanks blah, blah, blah.''
 \item[Dedication] Optional.
 \item[Subject class and Keywords] Normally present. Use
     only one primary subject class. There may be several
     secondary subject classes.
 \item[Abstract] The abstract should be self-contained; in
     particular, it should not include citations or
     references to the paper itself, such as
     ``in~\cite{Con}'' or ``in Theorem~\ref{T:main}".
\end{description}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Main matter}


%%%%%%%%%%%%%%%%%%%%%
\subsection{Structure}
Unless a paper is very short, it should be divided into
sections; the first section (Section~1 not Section~0)
immediately follows the abstract. Capitalization for section
titles follows the same rule as for paper titles; see
Section~\ref{S:Fm}.

%%%%%%%%%%%%%%%%%%%%%
\subsection{Formatting}
Use the standard \LaTeX\ environments to create your article's
formatting:
\begin{itemize}
 \item Lists should be created using the \verb+itemize+,
     \verb+enumerate+, or \verb+description+ environments.
     (You may use the \verb+list+ environment and the
     \texttt{enumerate} package for customized lists.)
 \item Declarations (theorem-type statements) and proofs
     should be formatted using the standard environments,
     as illustrated in Section~\ref{S:ex}.
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%
\subsection{Spacing}
As a rule, there should be {\bfseries no vertical spacing
commands} in your \LaTeX\ file; appropriate vertical spacings
are encoded in the \texttt{au} class file. Any final vertical
adjustments, including page breaks, are done by the editorial
office.

The \TeX\ system is designed to typeset mathematics with the
correct spacing. So, in general, you should not need to insert
tiny amounts of horizontal space into your mathematical
expressions. If the spacing in a mathematical expression does
not look right, it could be one of the following problems.
\begin{itemize}
 \item If you use a binary relation symbol in a
     non-standard way, then you may need to put braces
     around it to obtain the correct spacing. (For example,
     use $(0,1) \in {\le}$ rather than $(0,1) \in \le$.)
 \item If you use a non-standard binary relation or
     operation symbol, then use \verb+\mathrel+ or
     \verb+\mathbin+ to correct the spacing. (For example,
     use $x \mathrel{\theta} y$ and $a \mathbin{\hat+} b =
     c$ rather than $x \theta y$ and $a \hat+ b = c$).
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%
\subsection{References}
{\bfseries Do not hardwire} citations and internal references.
Use \verb+\cite+ for citations and use \verb+\label+ and
\verb+\ref+ for internal references, except that internal
references to equations should be done using \verb+\eqref+.

When referring by number to your figures, equations, theorems,
etc., capitalize and do not abbreviate. For example, use
\verb+Subsection~\ref{style}+ to refer to
Subsection~\ref{style}. Refer to external results as
in~\cite[Theorem 1]{Bet} or~\cite[p.~42]{Future}.

%%%%%%%%%%%%%%%%%%%%%
\subsection{Style}\label{style}
Definitions, formal or informal, should be set off in
\emph{emphatic mode} (not in \textbf{boldface}).

Pay attention to displaying mathematical formulas. While long
formulas (more than half a line long) should usually be \emph{displayed}, as in
\[
(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4,
\]
short formulas should usually be presented \emph{inline}, as in
$a + b = b + a$.

(Note that, in the \LaTeX\ file, the displayed formula has its
concluding punctuation mark \emph{inside} the delimiters, but
the inline formula has it \emph{outside} the delimiters.)

An article, even a mathematical one, should be written in
paragraphs. Please avoid line-breaking commands like \verb+\\+
in the text of your article.\footnote{Please \emph{do not} use
footnotes.}

%%%%%%%%%%%%%%%%%%%%%
\subsection{Line breaks}
Please ensure that your article does not have any lines that
stick out into the right margin. (These lines will be marked in
the right margin by a black `slug'.) You should be able to fix
these lines by
\begin{itemize}
 \item rephrasing a sentence,
 \item displaying a long formula, or
 \item adding optional hyphens (as in
     \verb+what\-cha\-ma\-call\-it+).
\end{itemize}
You may also take the opportunity to fix other unattractive
line breaks: for~instance, if an inline formula is being broken
at an inappropriate place. (For bad line breaks in text, the
`unbreakable space' \verb+~+ may be useful.)

%%%%%%%%%%%%%%%%%%%%%
\subsection{A few finer points}
\begin{enumerate}
 \item For a function $f$ from $A$ to $B$, write
     \verb+$f \colon A \to B$+. This is typeset as $f
     \colon A \to B$; note the nice spacing.
 \item If you use the command \verb+\dots+, then \LaTeX\
     will generally choose the correct position and spacing
     for your dots, as in
     \[
     1, 2, \dots, n \quad \text{and} \quad 1 + 2  + \dots + n.
     \]
 \item Mathematical operators should be declared in the
     preamble of the article. For example,
     \verb+\DeclareMathOperator{\dom}{dom}+. \emph{Then
     $\dom f$ will be typeset correctly, even in an
     italicized theorem statement.}
 \item Do not use the \verb+eqnarray+ environment. Please
     use \verb+align+ or \verb+alignat+ instead; see the
     proof of Theorem~\ref{T:main} for an example.
 \item Avoid using the default math font for words or
     abbreviations of words in formulas. For example, use
     $A_{\mathrm{fix}}$ rather than $A_{fix}$.
\end{enumerate}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Examples}\label{S:ex}

To illustrate some of the standard \LaTeX\ constructions, we
give a proof that the two-element Boolean algebra is primal.

\begin{definition}\label{D:bool}
We use $\B = \langle \{0,1\}; \vee, \wedge, {}', 0, 1\rangle$
to denote the \emph{two-element Boolean algebra}; see
Figure~\ref{F:bool}.
\end{definition}


%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht] %choose h,b,t (here, bottom, top)
%%%
\begin{tabular}{c|cc}
  $\vee$ & $0$ & $1$ \\\hline
  $0$ & $0$ & $1$ \\
  $1$ & $1$ & $1$
\end{tabular}
 \qquad
\begin{tabular}{c|cc}
  $\wedge$ & $0$ & $1$ \\\hline
  $0$ & $0$ & $0$ \\
  $1$ & $0$ & $1$
\end{tabular}
 \qquad
\begin{tabular}{c|c}
  & ${}'$ \\\hline
  $0$ & $1$ \\
  $1$ & $0$
\end{tabular}
%%%
%%% To include a separate file as a picture instead,
%%% you can use the graphics package and the command:
%\includegraphics[scale=1.0]{filename}
%%%
\caption{The operations of $\B$}\label{F:bool}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%


\begin{theorem}\label{T:main}
For all $n \in \N$, every function $f \colon \{0,1\}^n \to
\{0,1\}$ is an $n$-ary term function of\/~$\B$.
\end{theorem}

\begin{proof}
Let $f \colon \{0,1\}^n \to \{0,1\}$, for some $n \in \N$. We
start by introducing the notation
\[
x^i =
 \begin{cases}
 x', &\text{for $i = 0$;}\\
 x,  &\text{for $i = 1$.}
 \end{cases}
\]
Now define the term function $t \colon \{0,1\}^n \to \{0,1\}$
of $\B$ by
\begin{equation}
t(\vec x) = \bigvee \{\, x_1^{a_1} \wedge \dots \wedge
x_n^{a_n} \mid \vec a \in \inv f1 \,\}. \label{eq:dnf}
\end{equation}
For all $\vec b \in \{0,1\}^n$, it follows that
\begin{align*}
t(\vec b) = 1
 &\iff
b_1^{a_1} \wedge \dots \wedge b_n^{a_n} = 1, \text{ for some } \vec a \in \inv f1 \\
 &\iff
b_i^{a_i} = 1, \text{ for some } \vec a \in \inv f1 \text{ and all } i \in \{1,\dots,n\} \\
 &\iff
\vec b = \vec a, \text{ for some } \vec a \in \inv f1 \\
 &\iff
f(\vec b) = 1.
\end{align*}
Hence $f$ agrees with the term function~$t$, as required.
\end{proof}

\begin{remark}
In the proof of Theorem~\ref{T:main}, we showed how to write a
Boolean function in \emph{disjunctive normal form}; see
Equation~\eqref{eq:dnf}.
\end{remark}


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\section{Bibliography}

Algebra Universalis now uses the Springer `Math and Physical
Sciences' reference style. Articles in journals should be as
in~\cite{ABG,Bet,Con}; please follow carefully the format and
punctuation. Book chapters are as in~\cite{DE,Div,DG}, theses
as in~\cite{Foo}, and books as
in~\cite{Comp,Future,Sur75,Sur06}. For publications on the
internet, conference papers and manuscripts,
see~\cite{Short,Zed,Zed2}.


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%% Acknowledgments (Optional)
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\subsection*{Acknowledgment}
The editorial office of Algebra Universalis greatly appreciates
your effort in producing a \LaTeX\ file in the AU format.
Please remember that all our editors are volunteers, and your
cooperation eases their workload.


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%%  BIBLIOGRAPHY                                                 %%
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\begin{thebibliography}{99}

%% journal article
\bibitem{ABG}
    Alpha, C.I.,  Beta, B.,  Gamma, G.:
    On a problem of Dracula.
    Trans. Sylvanian Math. Soc.
    \textbf{18}, 455--460 (1987)


%% journal article, non-English
\bibitem{Bet}
    Beta, B.:
    The restricted Dracula problem.
    Izv. Akad. Nauk SSSR Ser. Mat.
    \textbf{23}, 142--157 (1988) (Russian)


%% journal article, in press
\bibitem{Con}
    Context, G.M.:
    A new concept.
    Algebra Universalis
    (in press)


%% book chapter, conference proceedings as volume in a series
\bibitem{DE}
    Delta, D., Epsilon, E.:
    Continuous functions.
    In:
    Topology and its Applications (Bra\c sov, 1999).
    Colloq. Sylvanian Math. Soc., vol. 21, pp. 23--50.
    South-Holland, Rotterdam (2000)


%% book chapter, names of editors
\bibitem{Div}
    Divad, D.:
    A survey of combinatorial palindromes.
    In:
    Kurtz, K., Lang, L. (eds.)
    The Short and Long of It, pp. 234--432.
    World Mathematics, New York (2008)


%% book chapter, conference proceedings as volume in a series
\bibitem{DG}
    Divad, D., Gardner, A.:
    Organic lattice growing.
    In:
    Proceedings of the First International Workshop on Natural Order (WNO 2000).
    Lecture Notes in Algebra, vol. 9, pp. 20--32.
    Winnebago University Press, Winnebago (2002)


%% thesis
\bibitem{Foo}
    Foo, S.-K.:
    Lattice Constructions.
    PhD thesis, University of Winnebago (1990)


%% book
\bibitem{More}
    Gr\"atzer, G.:
    More Math into LaTeX, 4th edn.
    Springer, New York (2007)


%% publication on the internet
\bibitem{Short}
    Gr\"atzer, G.:
    Short Course.
    \url{http://www.ctan.org/tex-archive/info/Math_into_LaTeX-4/}


%% book
\bibitem{Comp}
    Mittelbach, F.,  Goossens, M.,  Braams, J.,
    Carlisle, D.,  Rowley, C.:
    The LaTeX Companion, 2nd edn.
    Addison-Wesley, Boston (2004)


%% book
\bibitem{Future}
   Seer, H.A.:
   The Future of Algebra.
   D. Van Nostradamus, Winnebago (1968)


%% book volume
\bibitem{Sur75}
   Surname, F.M.:
   Introduction to Algebra, vol. 1.
   John Fox \& Sons, Melbourne (1975)


%% book, volume in a series
\bibitem{Sur06}
   Surname, F.M.:
   A Brief History of Algebra.
   Studies in Pure Mathematics, vol. 21.
   Winnebago University Press, Winnebago (2006)


%% paper presented at conference
\bibitem{Zed}
    Zed, Z.:
    Recent progress on the Dracula problem.
    Paper presented at the 3rd international conference on
    applied ordered sets, University of Winnebago, Winnebago, 2--5 June 2008


%% manuscript/preprint
\bibitem{Zed2}
    Zed, Z.:
    The number of lattices of size 17 (2009, preprint)

\end{thebibliography}
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\end{document}