From e0c6872cf40896c7be36b11dcc744620f10adf1d Mon Sep 17 00:00:00 2001 From: Norbert Preining Date: Mon, 2 Sep 2019 13:46:59 +0900 Subject: Initial commit --- info/examples/FirstSteps/Contents | 23 +++ info/examples/FirstSteps/article.tpl | 94 +++++++++++++ info/examples/FirstSteps/article2.tpl | 100 +++++++++++++ info/examples/FirstSteps/bibl.tpl | 135 ++++++++++++++++++ info/examples/FirstSteps/gallery.tex | 254 +++++++++++++++++++++++++++++++++ info/examples/FirstSteps/ggamsart.tpl | 112 +++++++++++++++ info/examples/FirstSteps/ggart.tpl | 97 +++++++++++++ info/examples/FirstSteps/ggart2.tpl | 101 +++++++++++++ info/examples/FirstSteps/intrart.tex | 127 +++++++++++++++++ info/examples/FirstSteps/lattice.sty | 122 ++++++++++++++++ info/examples/FirstSteps/math.tex | 21 +++ info/examples/FirstSteps/mathb.tex | 24 ++++ info/examples/FirstSteps/note1.tex | 22 +++ info/examples/FirstSteps/note1b.tex | 22 +++ info/examples/FirstSteps/note2.tex | 14 ++ info/examples/FirstSteps/noteslug.tex | 22 +++ info/examples/FirstSteps/sampart.tex | 258 ++++++++++++++++++++++++++++++++++ info/examples/FirstSteps/sampart2.tex | 252 +++++++++++++++++++++++++++++++++ 18 files changed, 1800 insertions(+) create mode 100644 info/examples/FirstSteps/Contents create mode 100644 info/examples/FirstSteps/article.tpl create mode 100644 info/examples/FirstSteps/article2.tpl create mode 100644 info/examples/FirstSteps/bibl.tpl create mode 100644 info/examples/FirstSteps/gallery.tex create mode 100644 info/examples/FirstSteps/ggamsart.tpl create mode 100644 info/examples/FirstSteps/ggart.tpl create mode 100644 info/examples/FirstSteps/ggart2.tpl create mode 100644 info/examples/FirstSteps/intrart.tex create mode 100644 info/examples/FirstSteps/lattice.sty create mode 100644 info/examples/FirstSteps/math.tex create mode 100644 info/examples/FirstSteps/mathb.tex create mode 100644 info/examples/FirstSteps/note1.tex create mode 100644 info/examples/FirstSteps/note1b.tex create mode 100644 info/examples/FirstSteps/note2.tex create mode 100644 info/examples/FirstSteps/noteslug.tex create mode 100644 info/examples/FirstSteps/sampart.tex create mode 100644 info/examples/FirstSteps/sampart2.tex (limited to 'info/examples/FirstSteps') diff --git a/info/examples/FirstSteps/Contents b/info/examples/FirstSteps/Contents new file mode 100644 index 0000000000..ccfa66eefb --- /dev/null +++ b/info/examples/FirstSteps/Contents @@ -0,0 +1,23 @@ +Sample articles and notes: + +gallery.tex the formulas from Chapter 3 +intrart.tex introductory sample article, Chapter 4 +lattice.sty personalized file for user defined commands, Section 5.6 +note1.tex first note, Section 1.3 +note1b.tex note1.tex with lines too long, Section 1.4 +noteslug.tex note1b.tex with slug, Section 1.4 +note2.tex second note, Section 1.5 +math.tex first note with math, Section 2.1 +mathb.tex first note with math with mistakes, Section 2.2 +sampart.tex sample article with AMS, Chapter 5 +sampart2.tex sample article with AMS and user defined commands, Section 5.6 + + +Templates: + +article.tpl article template for one author, Section 4.3 +article2.tpl article template for two authors, Section 4.3 +bibl.tpl templates for bibliographic entries, Section 4.4.4 +ggart.tpl article.tpl personalized, Section 4.3 +ggart2.tpl article2.tpl personalized, Section 4.3 +ggamsart.tpl personalized template for amsart document class, Section 5.4 diff --git a/info/examples/FirstSteps/article.tpl b/info/examples/FirstSteps/article.tpl new file mode 100644 index 0000000000..110b108671 --- /dev/null +++ b/info/examples/FirstSteps/article.tpl @@ -0,0 +1,94 @@ +% Sample file: article.tpl +% Typeset with LaTeX format + +\documentclass{article} +\usepackage{amssymb,latexsym,amsmath} + +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}{Lemma} +\newtheorem{proposition}{Proposition} +\newtheorem{definition}{Definition} +\newtheorem{corollary}{Corollary} +\newtheorem{notation}{Notation} + +\begin{document} +\title{titleline1\\ + titleline2} +\author{name\thanks{support}\\ + addressline1\\ + addressline2\\ + addressline3} +\date{date} +\maketitle + +\begin{abstract} + abstract text +\end{abstract} + +\begin{thebibliography}{99} + bibliographic entries +\end{thebibliography} +\end{document} + +Papers: + +\bibitem{xxx} + author, \emph{title,} journal \textbf{volume} + (year), pages. + +Books: + +\bibitem{xxx} + author, \emph{title,} publisher, address, year. + +\bibitem{xxx} + author, \emph{title,} series, vol.~volume, + publisher, address, edition, date. + +\bibitem{xxx} + editor, ed., \emph{title,} publisher, address, year. + +Papers in books: + +\bibitem{xxx} + author, \emph{title,} book title, publisher, + year, pp~pages. + +\bibitem{xxx} + author, \emph{title,} book title (editor, ed.), + vol.~volume, publisher, publisher address, date, + pp.~pages. + +Theses: + +\bibitem{xxx} + author, \emph{title,} Ph.D. thesis, university, year. + +Tech reports: + +\bibitem{xxx} + author, \emph{title,} tech. report, university, year. + +Research notes: + +\bibitem{xxx} + author, \emph{title,} Research Note number, + university, location, date, research paper in + preparation. + +Conference proceedings: + +\bibitem{xxx} + author, \emph{title,} conference title (location, + year). + +\bibitem{xxx} + author, \emph{title,} conference title, year + (editor, ed.), vol.~volume, publisher, address, + pp.~pages. + +Abstracts: + +\bibitem{xxx} + author, \emph{title,} Abstract: journal, volume, + year. \ No newline at end of file diff --git a/info/examples/FirstSteps/article2.tpl b/info/examples/FirstSteps/article2.tpl new file mode 100644 index 0000000000..3075530cd2 --- /dev/null +++ b/info/examples/FirstSteps/article2.tpl @@ -0,0 +1,100 @@ +% Sample file: article2.tpl for two authors +% Typeset with LaTeX format + +\documentclass{article} +\usepackage{amssymb,latexsym,amsmath} + +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}{Lemma} +\newtheorem{proposition}{Proposition} +\newtheorem{definition}{Definition} +\newtheorem{corollary}{Corollary} +\newtheorem{notation}{Notation} + +\begin{document} + +\title{titleline1\\ + titleline2} +\author{name1\thanks{support1}\\ + address1line1\\ + address1line2\\ + address1line3 + \and + name2\thanks{support2}\\ + address2line1\\ + address2line2\\ + address2line3} +\date{date} +\maketitle + +\begin{abstract} + abstract text +\end{abstract} + +\begin{thebibliography}{99} + bibliographic entries +\end{thebibliography} +\end{document} + +Papers: + +\bibitem{xxx} + author, \emph{title,} journal \textbf{volume} + (year), pages. + +Books: + +\bibitem{xxx} + author, \emph{title,} publisher, address, year. + +\bibitem{xxx} + author, \emph{title,} series, vol.~volume, + publisher, address, edition, date. + +\bibitem{xxx} + editor, ed., \emph{title,} publisher, address, year. + +Papers in books: + +\bibitem{xxx} + author, \emph{title,} book title, publisher, + year, pp~pages. + +\bibitem{xxx} + author, \emph{title,} book title (editor, ed.), + vol.~volume, publisher, publisher address, date, + pp.~pages. + +Theses: + +\bibitem{xxx} + author, \emph{title,} Ph.D. thesis, university, year. + +Tech reports: + +\bibitem{xxx} + author, \emph{title,} tech. report, university, year. + +Research notes: + +\bibitem{xxx} + author, \emph{title,} Research Note number, + university, location, date, research paper in + preparation. + +Conference proceedings: + +\bibitem{xxx} + author, \emph{title,} conference title (location, + year). + +\bibitem{xxx} + author, \emph{title,} conference title, year + (editor, ed.), vol.~volume, publisher, address, + pp.~pages. + +Abstracts: + +\bibitem{xxx} + author, \emph{title,} Abstract: journal, volume, + year. \ No newline at end of file diff --git a/info/examples/FirstSteps/bibl.tpl b/info/examples/FirstSteps/bibl.tpl new file mode 100644 index 0000000000..d7c40bb1e6 --- /dev/null +++ b/info/examples/FirstSteps/bibl.tpl @@ -0,0 +1,135 @@ +% Sample file: bibl.tpl +% bibliography template file + +\begin{thebibliography}{99} +\bibitem{hA70} + Henry~H. Albert, + \emph{Free torsoids,} + Current Trends in Lattice Theory, D.~Van Nostrand, 1970. +\bibitem{hA70a} + Henry~H. Albert, + \emph{Free torsoids,} + Current Trends in Lattice Theory (G.H. Birnbaum, ed.), vol.~7, + D.~Van Nostrand, Princeton-Toronto-London-Melbourne, + January 1970, + no translation available, pp.~173--215 (Hungarian). +\bibitem{sF90} + Soo-Key Foo, + \emph{Lattice Constructions,} + Ph.D. thesis, + University of Winnebago, 1990. +\bibitem{sF90a} + Soo-Key Foo, + \emph{Lattice Constructions,} + Ph.D. thesis, + University of Winnebago, Winnebago, MN, December, 1990, + final revision not yet available (Chinese). +\bibitem{gF86} + Grant~H. Foster, + \emph{Computational complexity in lattice theory,} + tech. report, Carnegie Mellon University, 1986. +\bibitem{gF86a} + Grant~H. Foster, + \emph{Computational complexity in lattice theory,} + Research Note 128A, Carnegie Mellon University, + Pittsburgh PA, December 1986, + research article in preparation (English). +\bibitem{pK69} + Peter~A. Konig, + \emph{Composition of functions,} + Proceedings of the Conference on Universal Algebra + (Kingston, 1969). +\bibitem{pK69a} + Peter~A. Konig, + \emph{Composition of functions,} + Proceedings of the Conference on Universal Algebra, 1969 + (G.H. Birnbaum, ed.), vol.~7, Canadian Mathematical + Society, Queen's Univ., available from the Montreal office, + pp.~1--106 (English). +\bibitem{wL75} + William~A. Landau, + \emph{Representations of complete lattices,} + Abstract: Notices Amer. Math. Soc., \textbf{18}, 937. +\bibitem{wL75a} + William~A. Landau, + \emph{Representations of complete lattices,} + Abstract: Notices Amer. Math. Soc. \textbf{18}, 937, + December 1975 (English). +\bibitem{gM68} + George~A. Menuhin, + \emph{Universal Algebra,} + D.~van Nostrand, Princeton-Toronto-London-Melbourne, 1968. +\bibitem{gM68a} + George~A. Menuhin, + \emph{Universal Algebra,} + University Series in Higher Mathematics, vol.~58, + D.~van Nostrand, Princeton-Toronto-London-Melbourne, + second ed., March 1968, no Russian translation (English). +\bibitem{eM57} + Ernest~T. Moynahan, + \emph{On a problem of M.H. Stone,} + Acta Math. Acad. Sci. Hungar. \textbf{8} (1957), 455--460. +\bibitem{eM57a} + Ernest~T. Moynahan, + \emph{On a problem of M.H. Stone,} + Acta Math. Acad. Sci. Hungar. \textbf{8} (1957), 455--460, + Russian translation available (English). +\end{thebibliography} + +Papers: + +\bibitem{xxx} +author, \emph{title,} journal \textbf{volume} (year), +pp.~pages. + +Books: + +\bibitem{xxx} +author, \emph{title,} publisher, address, year. + +\bibitem{xxx} +author, \emph{title,} series, vol.~volume, publisher, + address, edition, date. + +\bibitem{xxx} +editor, ed., \emph{title,} publisher, address, year. + +Papers in books: + +\bibitem{xxx} +author, \emph{title,} book title, publisher, year, pp~pages. + +\bibitem{xxx} +author, \emph{title,} book title (editor, ed.), vol.~volume, +publisher, publisher address, date, pp.~pages. + +Theses: + +\bibitem{xxx} +author, \emph{title,} Ph.D. thesis, university, year. + +Tech reports: + +\bibitem{xxx} +author, \emph{title,} tech. report, university, year. + +Research notes: + +\bibitem{xxx} +author, \emph{title,} Research Note number, university, +location, date, research paper in preparation. + +Conference proceedings: + +\bibitem{xxx} +author, \emph{title,} conference title (location, year). + +\bibitem{xxx} +author, \emph{title,} conference title, year + (editor, ed.), vol.~volume, publisher, address, + pp.~pages. + +Abstracts: + +\bibitem{xxx} +author, \emph{title,} Abstract: journal, volume, year. \ No newline at end of file diff --git a/info/examples/FirstSteps/gallery.tex b/info/examples/FirstSteps/gallery.tex new file mode 100644 index 0000000000..2dee9d3ff0 --- /dev/null +++ b/info/examples/FirstSteps/gallery.tex @@ -0,0 +1,254 @@ +% Sample file: gallery.tex formula template file +% Typeset with LaTeX format + +\documentclass{article} + +\usepackage{amssymb,latexsym,amsmath} + +\begin{document} + +Section 3.1 Formula gallery + +Formula 1 +\[ + x \mapsto \{\, c \in C \mid c \leq x \,\} +\] + +Formula 2 +\[ + \left| \bigcup (\, I_{j} \mid j \in J \,) \right| + < \mathfrak{m} +\] + +Formula 3 +\[ + A = \{\, x \in X \mid x \in X_{i}, + \mbox{ for some } i \in I \,\} +\] + +Formula 4 +\[ + \langle a_{1}, a_{2} \rangle \leq \langle a'_{1}, a'_{2}\rangle + \qquad \mbox{if{f}} \qquad a_{1} < a'_{1} \quad \mbox{or} + \quad a_{1} = a'_{1} \mbox{ and } a_{2} \leq a'_{2} +\] + +Formula 5 +\[ + \Gamma_{u'} = \{\, \gamma \mid \gamma < 2\chi, + \ B_{\alpha} \nsubseteq u', \ B_{\gamma} \subseteq u' \,\} +\] + +Formula 6 +\[ + A = B^{2} \times \mathbb{Z} +\] + +Formula 7 +\[ + \left( \bigvee (\, s_{i} \mid i \in I \,) \right)^{c} = + \bigwedge (\, s_{i}^{c} \mid i \in I \,) +\] + +Formula 8 +\[ + y \vee \bigvee (\, [B_{\gamma}] \mid \gamma + \in \Gamma \,) \equiv z \vee \bigvee (\, [B_{\gamma}] + \mid \gamma \in \Gamma \,) \pmod{ \Phi^{x} } +\] + +Formula 9 +\[ + f(\mathbf{x}) = \bigvee\nolimits_{\!\mathfrak{m}} + \left(\, + \bigwedge\nolimits_{\mathfrak{m}} + (\, x_{j} \mid j \in I_{i} \,) \mid i < \aleph_{\alpha} + \,\right) +\] + +Formula 10 +\[ + \left. \widehat{F}(x) \right|_{a}^{b} = + \widehat{F}(b) - \widehat{F}(a) +\] + +Formula 11 +\[ + u \underset{\alpha}{+} v \overset{1}{\thicksim} w + \overset{2}{\thicksim} z +\] + +Formula 12 +\[ + f(x) \overset{ \text{def} }{=} x^{2} - 1 +\] + +Formula 13 +\[ + \overbrace{a + b + \cdots + z}^{n} +\] + +Formula 14 +\[ + \begin{vmatrix} + a + b + c & uv\\ + a + b & c + d + \end{vmatrix} + = 7 +\] + +\[ + \begin{Vmatrix} + a + b + c & uv\\ + a + b & c + d + \end{Vmatrix} + = 7 +\] + +Formula 15 +\[ + \sum_{j \in \mathbf{N}} b_{ij} \hat{y}_{j} = + \sum_{j \in \mathbf{N}} b^{(\lambda)}_{ij} \hat{y}_{j} + + (b_{ii} - \lambda_{i}) \hat{y}_{i} \hat{y} +\] + +Formula 16 +\[ + \left( \prod^n_{\, j = 1} \hat x_{j} \right) H_{c} = + \frac{1}{2} \hat k_{ij} \det \hat{ \mathbf{K} }(i|i) +\] + +\[ + \biggl( \prod^n_{\, j = 1} \hat x_{j} \biggr) H_{c} = + \frac{1}{2} \hat{k}_{ij} \det \widehat{ \mathbf{K} }(i|i) +\] + +Formula 17 +\[ + \det \mathbf{K} (t = 1, t_{1}, \ldots, t_{n}) = + \sum_{I \in \mathbf{n} }(-1)^{|I|} + \prod_{i \in I} t_{i} + \prod_{j \in I} (D_{j} + \lambda_{j} t_{j}) + \det \mathbf{A}^{(\lambda)} (\,\overline{I} | \overline{I}\,) = 0 +\] + +Formula 18 +\[ + \lim_{(v, v') \to (0, 0)} + \frac{H(z + v) - H(z + v') - BH(z)(v - v')} + {\| v - v' \|} = 0 +\] + +Formula 19 +\[ + \int_{\mathcal{D}} | \overline{\partial u} |^{2} + \Phi_{0}(z) e^{\alpha |z|^2} \geq + c_{4} \alpha \int_{\mathcal{D}} |u|^{2} \Phi_{0} + e^{\alpha |z|^{2}} + c_{5} \delta^{-2} \int_{A} + |u|^{2} \Phi_{0} e^{\alpha |z|^{2}} +\] + +Formula 20 +\[ + \mathbf{A} = + \begin{pmatrix} + \dfrac{\varphi \cdot X_{n, 1}} + {\varphi_{1} \times \varepsilon_{1}} + & (x + \varepsilon_{2})^{2} & \cdots + & (x + \varepsilon_{n - 1})^{n - 1} + & (x + \varepsilon_{n})^{n}\\[10pt] + \dfrac{\varphi \cdot X_{n, 1}} + {\varphi_{2} \times \varepsilon_{1}} + & \dfrac{\varphi \cdot X_{n, 2}} + {\varphi_{2} \times \varepsilon_{2}} + & \cdots & (x + \varepsilon_{n - 1})^{n - 1} + & (x + \varepsilon_{n})^{n}\\ + \hdotsfor{5}\\ + \dfrac{\varphi \cdot X_{n, 1}} + {\varphi_{n} \times \varepsilon_{1}} + & \dfrac{\varphi \cdot X_{n, 2}} + {\varphi_{n} \times \varepsilon_{2}} + & \cdots & \dfrac{\varphi \cdot X_{n, n - 1}} + {\varphi_{n} \times \varepsilon_{n - 1}} + & \dfrac{\varphi\cdot X_{n, n}} + {\varphi_{n} \times \varepsilon_{n}} + \end{pmatrix} + + \mathbf{I}_{n} +\] + + +Section 3.2. User-defined commands + +Formula 20 with user-defined commands: + +\newcommand{\quot}[2]{% +\dfrac{\varphi \cdot X_{n, #1}}% +{\varphi_{#2} \times \varepsilon_{#1}}} +\newcommand{\exn}[1]{(x+\varepsilon_{#1})^{#1}} + +\[ + \mathbf{A} = + \begin{pmatrix} + \quot{1}{1} & \exn{2} & \cdots & \exn{n - 1}&\exn{n}\\[10pt] + \quot{1}{2} & \quot{2}{2} & \cdots & \exn{n - 1} &\exn{n}\\ + \hdotsfor{5}\\ + \quot{1}{n} & \quot{2}{n} & \cdots & + \quot{n - 1}{n} & \quot{n}{n} + \end{pmatrix} + + \mathbf{I}_{n} +\] + +Section 3.3. Building a formula step-by-step + +Step 1 +$\left[ \frac{n}{2} \right]$ + +Step 2 +\[ + \sum_{i = 1}^{ \left[ \frac{n}{2} \right] } +\] + +Step 3 +\[ + x_{i, i + 1}^{i^{2}} \qquad \left[ \frac{i + 3}{3} \right] +\] + +Step 4 +\[ + \binom{ x_{i,i + 1}^{i^{2}} }{ \left[ \frac{i + 3}{3} \right] } +\] + +Step 5 +$\sqrt{ \mu(i)^{ \frac{3}{2} } (i^{2} - 1) }$ + +$\sqrt{ \mu(i)^{ \frac{3}{2} } (i^{2} - 1) }$ + +Step 6 +$\sqrt[3]{ \rho(i) - 2 }$ $\sqrt[3]{ \rho(i) - 1 }$ + +Step 7 +\[ + \frac{ \sqrt{ \mu(i)^{ \frac{3}{2}} (i^{2} -1) } } + { \sqrt[3]{\rho(i) - 2} + \sqrt[3]{\rho(i) - 1} } +\] + +Step 8 +\[ + \sum_{i = 1}^{ \left[ \frac{n}{2} \right] } + \binom{ x_{i, i + 1}^{i^{2}} } + { \left[ \frac{i + 3}{3} \right] } + \frac{ \sqrt{ \mu(i)^{ \frac{3}{2}} (i^{2} - 1) } } + { \sqrt[3]{\rho(i) - 2} + \sqrt[3]{\rho(i) - 1} } +\] + +\[\sum_{i=1}^{\left[\frac{n}{2}\right]}\binom{x_{i,i+1}^{i^{2}}} +{\left[\frac{i+3}{3}\right]}\frac{\sqrt{\mu(i)^{\frac{3} +{2}}(i^{2}-1)}}{\sqrt[3]{\rho(i)-2}+\sqrt[3]{\rho(i)-1}}\] + +%\[\sum_{i=1}^{\left[\frac{n}{2}\right]}\binom{x_{i,i+1}^{i^{2}}} +%{\left[\frac{i+3}{3}\right]}\frac{\sqrt{\mu(i)^{\frac{3} +%{2}}}(i^{2}-1)}}{\sqrt[3]{\rho(i)-2}+\sqrt[3]{\rho(i)-1}}\] + + +\end{document} + diff --git a/info/examples/FirstSteps/ggamsart.tpl b/info/examples/FirstSteps/ggamsart.tpl new file mode 100644 index 0000000000..1f4a045881 --- /dev/null +++ b/info/examples/FirstSteps/ggamsart.tpl @@ -0,0 +1,112 @@ +% Sample file: ggams.tpl +% Typeset with LaTeX format + +\documentclass{amsart} +\usepackage{amssymb,latexsym} + +% theorems, corollaries, lemmas, and propositions, +% in the most emphatic (plain) style; +% all are numbered separately +% There is a Main Theorem in the most emphatic (plain) +% style, unnumbered +% There are definitions, in the less emphatic (definition) style +% There are notations, in the least emphatic (remark) style, +% unnumbered + +\theoremstyle{plain} +\newtheorem{theorem}{Theorem} +\newtheorem{corollary}{Corollary} +\newtheorem*{main}{Main Theorem} +\newtheorem{lemma}{Lemma} +\newtheorem{proposition}{Proposition} + +\theoremstyle{definition} +\newtheorem{definition}{Definition} + +\theoremstyle{remark} +\newtheorem*{notation}{Notation} + +\begin{document} +\title[short title]{titleline1\\ + titleline2} +\author{George Gr\"{a}tzer} +\address{University of Manitoba\\ + Department of Mathematics\\ + Winnipeg, MN, R3T 2N2\\ + Canada} +\email{gratzer@cc.umanitoba.ca} +\urladdr{http://server.maths.umanitoba.ca/homepages/gratzer/} +\thanks{Research supported by the NSERC of Canada} + +\keywords{keywords} +\subjclass{Primary: subject; Secondary: subject} +\date{date} + +\begin{abstract} + abstract text +\end{abstract} +\maketitle + +% Bibliography +\begin{thebibliography}{99} + bibliographic entries +\end{thebibliography} +\end{document} + +Papers: + +\bibitem{xxx} +author, \emph{title,} journal \textbf{volume} (year), +pp.~pages. + +Books: + +\bibitem{xxx} +author, \emph{title,} publisher, address, year. + +\bibitem{xxx} +author, \emph{title,} series, vol.~volume, publisher, + address, edition, date. + +\bibitem{xxx} +editor, ed., \emph{title,} publisher, address, year. + +Papers in books: + +\bibitem{xxx} +author, \emph{title,} book title, publisher, year, pp~pages. + +\bibitem{xxx} +author, \emph{title,} book title (editor, ed.), vol.~volume, +publisher, publisher address, date, pp.~pages. + +Theses: + +\bibitem{xxx} +author, \emph{title,} Ph.D. thesis, university, year. + +Tech reports: + +\bibitem{xxx} +author, \emph{title,} tech. report, university, year. + +Research notes: + +\bibitem{xxx} +author,\emph{title,} Research Note number, university, +location, date, research paper in preparation. + +Conference proceedings: + +\bibitem{xxx} +author, \emph{title,} conference title (location, year). + +\bibitem{xxx} +author, \emph{title,} conference title, year + (editor, ed.), vol.~volume, publisher, address, + pp.~pages. + +Abstracts: + +\bibitem{xxx} +author, \emph{title,} Abstract: journal, volume, year. \ No newline at end of file diff --git a/info/examples/FirstSteps/ggart.tpl b/info/examples/FirstSteps/ggart.tpl new file mode 100644 index 0000000000..15bd6718ed --- /dev/null +++ b/info/examples/FirstSteps/ggart.tpl @@ -0,0 +1,97 @@ +% Sample file: ggart.tpl +% Typeset with LaTeX format + +\documentclass{article} +\usepackage{amssymb,latexsym,amsmath} + +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}{Lemma} +\newtheorem{proposition}{Proposition} +\newtheorem{definition}{Definition} +\newtheorem{corollary}{Corollary} +\newtheorem{notation}{Notation} + +\begin{document} +\title{titleline1\\ + titleline2} +\author{George Gr\"{a}tzer\thanks{Research supported by the + NSERC of Canada.}\\ + University of Manitoba\\ + Department of Mathematics\\ + Winnipeg, MN, R3T 2N2\\ + Canada} +\date{date} +\maketitle + +\begin{abstract} + abstract +\end{abstract} + +\begin{thebibliography}{99} + + +\end{thebibliography} +\end{document} + +Papers: + +\bibitem{xxx} + author, \emph{title,} journal \textbf{volume} + (year), pages. + +Books: + +\bibitem{xxx} + author, \emph{title,} publisher, address, year. + +\bibitem{xxx} + author, \emph{title,} series, vol.~volume, + publisher, address, edition, date. + +\bibitem{xxx} + editor, ed., \emph{title,} publisher, address, year. + +Papers in books: + +\bibitem{xxx} + author, \emph{title,} book title, publisher, + year, pp~pages. + +\bibitem{xxx} + author, \emph{title,} book title (editor, ed.), + vol.~volume, publisher, publisher address, date, + pp.~pages. + +Theses: + +\bibitem{xxx} + author, \emph{title,} Ph.D. thesis, university, year. + +Tech reports: + +\bibitem{xxx} + author, \emph{title,} tech. report, university, year. + +Research notes: + +\bibitem{xxx} + author, \emph{title,} Research Note number, + university, location, date, research paper in + preparation. + +Conference proceedings: + +\bibitem{xxx} + author, \emph{title,} conference title (location, + year). + +\bibitem{xxx} + author, \emph{title,} conference title, year + (editor, ed.), vol.~volume, publisher, address, + pp.~pages. + +Abstracts: + +\bibitem{xxx} + author, \emph{title,} Abstract: journal, volume, + year. \ No newline at end of file diff --git a/info/examples/FirstSteps/ggart2.tpl b/info/examples/FirstSteps/ggart2.tpl new file mode 100644 index 0000000000..f56c2f781c --- /dev/null +++ b/info/examples/FirstSteps/ggart2.tpl @@ -0,0 +1,101 @@ +% Sample file: ggart2.tpl for two authors +% Typeset with LaTeX format + +\documentclass{article} +\usepackage{amssymb,latexsym,amsmath} + +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}{Lemma} +\newtheorem{proposition}{Proposition} +\newtheorem{definition}{Definition} +\newtheorem{corollary}{Corollary} +\newtheorem{notation}{Notation} + +\begin{document} +\title{titleline1\\ + titleline2} +\author{George Gr\"{a}tzer\thanks{Research supported by the + NSERC of Canada.}\\ + University of Manitoba\\ + Department of Mathematics\\ + Winnipeg, MN, R3T 2N2\\ + Canada + \and + name2\thanks{support2}\\ + address2line1\\ + address2line2\\ + address2line3} +\date{date} +\maketitle + +\begin{abstract} + abstract +\end{abstract} + +\begin{thebibliography}{99} + bibliographic entries +\end{thebibliography} +\end{document} + +Papers: + +\bibitem{xxx} + author, \emph{title,} journal \textbf{volume} + (year), pages. + +Books: + +\bibitem{xxx} + author, \emph{title,} publisher, address, year. + +\bibitem{xxx} + author, \emph{title,} series, vol.~volume, + publisher, address, edition, date. + +\bibitem{xxx} + editor, ed., \emph{title,} publisher, address, year. + +Papers in books: + +\bibitem{xxx} + author, \emph{title,} book title, publisher, + year, pp~pages. + +\bibitem{xxx} + author, \emph{title,} book title (editor, ed.), + vol.~volume, publisher, publisher address, date, + pp.~pages. + +Theses: + +\bibitem{xxx} + author, \emph{title,} Ph.D. thesis, university, year. + +Tech reports: + +\bibitem{xxx} + author, \emph{title,} tech. report, university, year. + +Research notes: + +\bibitem{xxx} + author, \emph{title,} Research Note number, + university, location, date, research paper in + preparation. + +Conference proceedings: + +\bibitem{xxx} + author, \emph{title,} conference title (location, + year). + +\bibitem{xxx} + author, \emph{title,} conference title, year + (editor, ed.), vol.~volume, publisher, address, + pp.~pages. + +Abstracts: + +\bibitem{xxx} + author, \emph{title,} Abstract: journal, volume, + year. \ No newline at end of file diff --git a/info/examples/FirstSteps/intrart.tex b/info/examples/FirstSteps/intrart.tex new file mode 100644 index 0000000000..1f35c47723 --- /dev/null +++ b/info/examples/FirstSteps/intrart.tex @@ -0,0 +1,127 @@ +% Introductory sample article: intrart.tex +% Typeset with LaTeX format + +\documentclass{article} +\usepackage{latexsym} +\newtheorem{theorem}{Theorem} +\newtheorem{definition}{Definition} +\newtheorem{notation}{Notation} + +\begin{document} +\title{A construction of complete-simple\\ + distributive lattices} +\author{George~A. Menuhin\thanks{Research supported + by the NSF under grant number~23466.}\\ + Computer Science Department\\ + Winnebago, Minnesota 23714\\ + menuhin@cc.uwinnebago.edu} +\date{March 15, 1999} +\maketitle + +\begin{abstract} + In this note, we prove that there exist \emph{complete-simple + distributive lattices,} that is, complete distributive + lattices in which there are only two complete congruences. +\end{abstract} + +\section{Introduction}\label{S:intro} +In this note, we prove the following result: + +\begin{theorem} + There exists an infinite complete distributive lattice $K$ + with only the two trivial complete congruence relations. +\end{theorem} + +\section{The $\Pi^{*}$ construction}\label{S:P*} +The following construction is crucial in the proof of our Theorem: + +\begin{definition}\label{D:P*} + Let $D_{i}$, for $i \in I$, be complete distributive + lattices satisfying condition~\textup{(J)}. Their + $\Pi^{*}$ product is defined as follows: + \[ + \Pi^{*} ( D_{i} \mid i \in I ) = + \Pi ( D_{i}^{-} \mid i \in I ) + 1; + \] + that is, $\Pi^{*} ( D_{i} \mid i \in I )$ is + $\Pi ( D_{i}^{-} \mid i \in I )$ with a new unit element. +\end{definition} + +\begin{notation} + If $i \in I$ and $d \in D_{i}^{-}$, then + \[ + \langle \ldots, 0, \ldots, d, \ldots, 0, \ldots \rangle + \] + is the element of $\Pi^{*} ( D_{i} \mid i \in I )$ whose + $i$-th component is $d$ and all the other components + are $0$. +\end{notation} + +See also Ernest~T. Moynahan~\cite{eM57a}. + +Next we verify the following result: + +\begin{theorem}\label{T:P*} + Let $D_{i}$, $i \in I$, be complete distributive + lattices satisfying condition~\textup{(J)}. Let $\Theta$ + be a complete congruence relation on + $\Pi^{*} ( D_{i} \mid i \in I )$. + If there exist $i \in I$ and $d \in D_{i}$ with + $d < 1_{i}$ such that, for all $d \leq c < 1_{i}$, + \begin{equation}\label{E:cong1} + \langle \ldots, d, \ldots, 0, \ldots \rangle \equiv + \langle \ldots, c, \ldots, 0, \ldots \rangle \pmod{\Theta}, + \end{equation} + then $\Theta = \iota$. +\end{theorem} + +\emph{Proof.} Since +\begin{equation}\label{E:cong2} + \langle \ldots, d, \ldots, 0, \ldots \rangle \equiv + \langle \ldots, c, \ldots, 0, \ldots \rangle \pmod{\Theta}, +\end{equation} +and $\Theta$ is a complete congruence relation, it follows +from condition~(J) that +\begin{equation}\label{E:cong} + \langle \ldots, d, \ldots, 0, \ldots \rangle \equiv + \bigvee ( \langle \ldots, c, \ldots, 0, \ldots \rangle + \mid d \leq c < 1 ) \pmod{\Theta}. +\end{equation} + +Let $j \in I$, $j \neq i$, and let $a \in D_{j}^{-}$. +Meeting both sides of the congruence (\ref{E:cong2}) with +$\langle \ldots, a, \ldots, 0, \ldots \rangle$, we obtain that +\begin{equation}\label{E:comp} + 0 = \langle \ldots, a, \ldots, 0, \ldots \rangle \pmod{\Theta}, +\end{equation} +Using the completeness of $\Theta$ and (\ref{E:comp}), +we get: +\[ + 0 \equiv \bigvee ( \langle \ldots, a, \ldots, 0, \ldots \rangle + \mid a \in D_{j}^{-} ) = 1 \pmod{\Theta}, +\] +hence $\Theta = \iota$. + +\begin{thebibliography}{9} + \bibitem{sF90} + Soo-Key Foo, + \emph{Lattice Constructions,} + Ph.D. thesis, + University of Winnebago, Winnebago, MN, December, 1990. + \bibitem{gM68} + George~A. Menuhin, + \emph{Universal Algebra,} + D.~van Nostrand, Princeton-Toronto-London-Melbourne, 1968. + \bibitem{eM57} + Ernest~T. Moynahan, + \emph{On a problem of M.H. Stone,} + Acta Math. Acad. Sci. Hungar. \textbf{8} (1957), 455--460. + \bibitem{eM57a} + Ernest~T. Moynahan, + \emph{Ideals and congruence relations in lattices.~II,} + Magyar Tud. Akad. Mat. Fiz. Oszt. K\"{o}zl. \textbf{9} + (1957), 417--434. +\end{thebibliography} + +\end{document} + diff --git a/info/examples/FirstSteps/lattice.sty b/info/examples/FirstSteps/lattice.sty new file mode 100644 index 0000000000..2ebb04e90e --- /dev/null +++ b/info/examples/FirstSteps/lattice.sty @@ -0,0 +1,122 @@ +% lattice.sty +% Command file for lattice papers +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{lattice} + [1999/03/15 Commands for lattices, First Steps] +\RequirePackage{amsmath} +\RequirePackage{amssymb} +\RequirePackage{latexsym} +\RequirePackage{eucal} + +% Lattice operations +\newcommand{\jj}{\vee}% join +\newcommand{\mm}{\wedge}% meet +\newcommand{\JJ}{\bigvee}% big join +\newcommand{\MM}{\bigwedge}% big meet +\newcommand{\JJm}[2]{\JJ(\,#1\mid#2\,)}% big join with a middle +\newcommand{\MMm}[2]{\MM(\,#1\mid#2\,)}% big meet with a middle + +% Set operations +\newcommand{\uu}{\cup}% union +\newcommand{\ii}{\cap}% intersection +\newcommand{\UU}{\bigcup}% big union +\newcommand{\II}{\bigcap}% big intersection +\newcommand{\UUm}[2]{\UU(\,#1\mid#2\,)}% big union with a middle +\newcommand{\IIm}[2]{\II(\,#1\mid#2\,)}% big intersection with a middle + +% Sets +\newcommand{\ci}{\subseteq}% contained in with equality +\newcommand{\nc}{\nsubseteq}% not \ci +\newcommand{\sci}{\subset}% strictly contained in with equality +\newcommand{\nci}{\nc}% not \ci +\newcommand{\ce}{\supseteq}% containing with equality +\newcommand{\nce}{\nsupseteq}% not \ce +\newcommand{\nin}{\notin}% not \in +\newcommand{\es}{\varnothing}% the empty set +\newcommand{\set}[1]{\{#1\}}% set +\newcommand{\setm}[2]{\{\,#1\mid#2\,\}}% set with a middle +\def\vv<#1>{\langle#1\rangle}% vector + +% Partial ordering +\newcommand{\nle}{\nleq}% not \leq + +% Greek letters +\newcommand{\ga}{\alpha} +\newcommand{\gb}{\beta} +\newcommand{\gc}{\chi} +\newcommand{\gd}{\delta} +\renewcommand{\ge}{\varepsilon}% use \geq for >= +\newcommand{\gf}{\varphi} +\renewcommand{\gg}{\gamma}% old use >> +\newcommand{\gh}{\eta} +\newcommand{\gi}{\iota} +\newcommand{\gj}{\theta} +\newcommand{\gk}{\kappa} +\newcommand{\gl}{\lambda} +\newcommand{\gm}{\mu} +\newcommand{\gn}{\nu} +\newcommand{\go}{\omega} +\newcommand{\gp}{\pi} +\newcommand{\gq}{\theta} +\newcommand{\gr}{\varrho} +\newcommand{\gs}{\sigma} +\newcommand{\gt}{\tau} +\newcommand{\gu}{\upsilon} +\newcommand{\gv}{\vartheta} +\newcommand{\gw}{\omega} +\newcommand{\gx}{\xi} +\newcommand{\gy}{\psi} +\newcommand{\gz}{\zeta} + +\newcommand{\gC}{\Xi} +\newcommand{\gG}{\Gamma} +\newcommand{\gD}{\Delta} +\newcommand{\gF}{\Phi} +\newcommand{\gL}{\Lambda} +\newcommand{\gO}{\Omega} +\newcommand{\gP}{\Pi} +\newcommand{\gQ}{\Theta} +\newcommand{\gS}{\Sigma} +\newcommand{\gU}{\Upsilon} +\newcommand{\gW}{\Omega} +\newcommand{\gX}{\Xi} +\newcommand{\gY}{\Psi} + +% Font commands +\newcommand{\tbf}{\textbf}% text bold +\newcommand{\tit}{\textit}% text italic +\newcommand{\tsl}{\textsl}% text slanted +\newcommand{\tsc}{\textsc}% text small cap +\newcommand{\ttt}{\texttt}% text typewriter +\newcommand{\trm}{\textrm}% text roman +\newcommand{\tsf}{\textsf}% text sans serif +\newcommand{\tup}{\textup}% text upright + +\newcommand{\mbf}{\mathbf}% math bold +\newcommand{\mit}{\mathit}% math italic +\newcommand{\msf}{\mathsf}% math sans serif +\newcommand{\mrm}{\mathrm}% math roman +\newcommand{\mtt}{\mathtt}% math typewriter + +\newcommand{\B}{\boldsymbol} + % Bold math symbol, use as \B{a} +\DeclareMathAlphabet{\Bi}{OT1}{cmm}{b}{it} + % Bold math italic, use as \Bi{a} +\newcommand{\C}[1]{\mathcal{#1}} + % Euler Script - only caps, use as \C{A} +\newcommand{\D}[1]{\mathbb{#1}} + % Doubled - blackboard bold - only caps, use as \D{A} +\newcommand{\E}[1]{\mathcal{#1}}% same as \C + % Euler Script - only caps, use as \E{A} +\newcommand{\F}[1]{\mathfrak{#1}}% Fraktur, use as \F{a} + +% Miscellaneous +\newcommand{\nl}{\newline} +\newcommand{\ol}[1]{\overline{#1}} +\newcommand{\ul}[1]{\underline{#1}} +\providecommand{\bysame}{\makebox[3em]{\hrulefill}\thinspace} +\newcommand{\q}{\quad}% spacing +\newcommand{\qq}{\qquad}% more spacing +\newcommand{\iso}{\cong}% isomorphic + +\endinput \ No newline at end of file diff --git a/info/examples/FirstSteps/math.tex b/info/examples/FirstSteps/math.tex new file mode 100644 index 0000000000..fc96b999eb --- /dev/null +++ b/info/examples/FirstSteps/math.tex @@ -0,0 +1,21 @@ +% Sample file: math.tex +% Typeset with LaTeX format +\documentclass{article} + +\begin{document} +In first-year calculus, we define intervals such as +$(u, v)$ and $(u, \infty)$. Such an interval is a +\emph{neighborhood} of $a$ +if $a$ is in the interval. Students should +realize that $\infty$ is only a +symbol, not a number. This is important since +we soon introduce concepts + such as $\lim_{x \to \infty} f(x)$. + +When we introduce the derivative, +\[ + \lim_{x \to a} \frac{f(x) - f(a)}{x - a}, +\] +we assume that the function is defined and continuous +in a neighborhood of $a$. +\end{document} diff --git a/info/examples/FirstSteps/mathb.tex b/info/examples/FirstSteps/mathb.tex new file mode 100644 index 0000000000..be97c54155 --- /dev/null +++ b/info/examples/FirstSteps/mathb.tex @@ -0,0 +1,24 @@ +% Sample file: mathB.tex +% Typeset with LaTeX format +\documentclass{article} + +\begin{document} +In first-year calculus, we define intervals such as +%$(u, v)$ and $(u, \infty)$. Such an interval is a + $(u, v)$ and (u, \infty)$. Such an interval is a + \emph{neighborhood} of $a$ +if $a$ is in the interval. Students should +realize that $\infty$ is only a +symbol, not a number. This is important since +we soon introduce concepts + such as $\lim_{x \to \infty} f(x)$. +%such as $\lim_{x \to \infty f(x)$. + +When we introduce the derivative, +\[ + \lim_{x \to a} \frac{f(x) - f(a)}{x - a} + %\lim_{x \to a} \frac{f(x) - f(a) x - a} +\] +we assume that the function is defined and continuous +in a neighborhood of $a$. +\end{document} diff --git a/info/examples/FirstSteps/note1.tex b/info/examples/FirstSteps/note1.tex new file mode 100644 index 0000000000..e4425f6779 --- /dev/null +++ b/info/examples/FirstSteps/note1.tex @@ -0,0 +1,22 @@ +% Sample file: note1.tex +% Typeset with LaTeX format +\documentclass{article} + +\begin{document} +It is of some concern to me that +the terminology used in multi-section + math courses is not uniform. + +In several sections of the course on +matrix theory, the term + ``hamiltonian-reduced'' is used. + I, personally, would rather call these ``hyper-simple.'' I +invite others to comment on this problem. + +Of special concern to me is the terminology in the course +by Prof.~Rudi Hochschwabauer. + Since his field is new, there is + no accepted +terminology. It is imperative +that we arrive at a satisfactory solution. +\end{document} diff --git a/info/examples/FirstSteps/note1b.tex b/info/examples/FirstSteps/note1b.tex new file mode 100644 index 0000000000..4de05f5a2b --- /dev/null +++ b/info/examples/FirstSteps/note1b.tex @@ -0,0 +1,22 @@ +% Sample file: note1b.tex +% Typeset with LaTeX format +\documentclass{article} + +\begin{document} +It is of some concern to me that +the terminology used in multi-section + math courses is not uniform. + +In several sections of the course on +matrix theory, the strange term + ``hamiltonian-reduced'' is used. + I, personally, would rather call these ``hyper-simple.'' I +invite others to comment on this problem. + +Of special concern to me is the terminology in the course +by Prof.~Hochschwabauer. + Since his field is new, there is + no accepted +terminology. It is imperative +that we arrive at a satisfactory solution. +\end{document} \ No newline at end of file diff --git a/info/examples/FirstSteps/note2.tex b/info/examples/FirstSteps/note2.tex new file mode 100644 index 0000000000..7b218f21fa --- /dev/null +++ b/info/examples/FirstSteps/note2.tex @@ -0,0 +1,14 @@ +% Sample file: note2.tex +% Typeset with LaTeX format +\documentclass{article} + +\begin{document} +\begin{flushright} + \today +\end{flushright} +\textbf{From the desk of George Gr\"{a}tzer}\\[22pt] +April~7--21 \emph{please} use my temporary e-mail address: +\begin{center} + \texttt{George\_Gratzer@umanitoba.ca} +\end{center} +\end{document} diff --git a/info/examples/FirstSteps/noteslug.tex b/info/examples/FirstSteps/noteslug.tex new file mode 100644 index 0000000000..04d2f46307 --- /dev/null +++ b/info/examples/FirstSteps/noteslug.tex @@ -0,0 +1,22 @@ +% Sample file: noteslug.tex +% Typeset with the LaTeX format +\documentclass[draft]{article} + +\begin{document} +It is of some concern to me that +the terminology used in multi-section + math courses is not uniform. + +In several sections of the course on +matrix theory, the strange term + ``hamiltonian-reduced'' is used. + I, personally, would rather call these ``hyper-simple.'' I +invite others to comment on this problem. + +Of special concern to me is the terminology in the course +by Prof.~Hochschwabauer. + Since his field is new, there is + no accepted +terminology. It is imperative +that we arrive at a satisfactory solution. +\end{document} \ No newline at end of file diff --git a/info/examples/FirstSteps/sampart.tex b/info/examples/FirstSteps/sampart.tex new file mode 100644 index 0000000000..1186358245 --- /dev/null +++ b/info/examples/FirstSteps/sampart.tex @@ -0,0 +1,258 @@ +% Sample file: sampart.tex +% The sample article for the amsart document class +% Typeset with LaTeX format + +\documentclass{amsart} +\usepackage{amssymb,latexsym} + +\theoremstyle{plain} +\newtheorem{theorem}{Theorem} +\newtheorem{corollary}{Corollary} +\newtheorem*{main}{Main~Theorem} +\newtheorem{lemma}{Lemma} +\newtheorem{proposition}{Proposition} + +\theoremstyle{definition} +\newtheorem{definition}{Definition} + +\theoremstyle{remark} +\newtheorem*{notation}{Notation} + +\numberwithin{equation}{section} + +\begin{document} +\title[Complete-simple distributive lattices] + {A construction of complete-simple\\ + distributive lattices} +\author{George~A. Menuhin} +\address{Computer Science Department\\ + University of Winnebago\\ + Winnebago, Minnesota 53714} +\email{menuhin@ccw.uwinnebago.edu} +\urladdr{http://math.uwinnebago.ca/homepages/menuhin/} +\thanks{Research supported by the NSF under grant number +~23466.} +\keywords{Complete lattice, distributive lattice, +complete congruence, + congruence lattice} +\subjclass{Primary: 06B10; Secondary: 06D05} +\date{March 15, 1999} +\begin{abstract} + In this note we prove that there exist \emph{complete-simple distributive + lattices,} that is, complete distributive lattices in which there are + only two complete congruences. +\end{abstract} + +\maketitle + +\section{Introduction}\label{S:intro} +In this note we prove the following result: + +\begin{main} + There exists an infinite complete distributive lattice $K$ with only + the two trivial complete congruence relations. +\end{main} + +\section{The $D^{\langle 2 \rangle}$ construction}\label{S:Ds} +For the basic notation in lattice theory and universal algebra, see Ferenc~R. +Richardson~\cite{fR82} and George~A. Menuhin~\cite{gM68}. We start with some +definitions: + +\begin{definition}\label{D:prime} + Let $V$ be a complete lattice, and let $\mathfrak{p} = [u, v]$ be + an interval of $V$. Then $\mathfrak{p}$ is called + \emph{complete-prime} if the following three conditions are satisfied: + \begin{itemize} + \item[(1)] $u$ is meet-irreducible but $u$ is \emph{not} + completely meet-irreducible; + \item[(2)] $v$ is join-irreducible but $v$ is \emph{not} + completely join-irreducible; + \item[(3)] $[u, v]$ is a complete-simple lattice. + \end{itemize} +\end{definition} + +Now we prove the following result: + +\begin{lemma}\label{L:ds} + Let $D$ be a complete distributive lattice satisfying + conditions~\textup{(1)} and~\textup{(2)}. Then + $D^{\langle 2 \rangle}$ is a sublattice of $D^{2}$; + hence $D^{\langle 2 \rangle}$ is a lattice, and + $D^{\langle 2 \rangle}$ is a complete distributive + lattice satisfying conditions~\textup{(1)} and \textup{(2)}. +\end{lemma} + +\begin{proof} + By conditions~(1) and (2), $D^{\langle 2 \rangle}$ is a sublattice + of $D^{2}$. Hence, $D^{\langle 2 \rangle}$ is a lattice. + + Since $D^{\langle 2 \rangle}$ is a sublattice of a distributive + lattice, $D^{\langle 2 \rangle}$ is a distributive lattice. Using + the characterization of standard ideals in Ernest~T. Moynahan~\cite{eM57}, + $D^{\langle 2 \rangle}$ has a zero and a unit element, + namely, $\langle 0, 0 \rangle$ and $\langle 1, 1 \rangle$. + To show that $D^{\langle 2 \rangle}$ is complete, let + $\varnothing \ne A \subseteq D^{\langle 2 \rangle}$, and let + $a = \bigvee A$ in $D^{2}$. If + $a \in D^{\langle 2 \rangle}$, then + $a = \bigvee A$ in $D^{\langle 2 \rangle}$; otherwise, $a$ + is of the form $\langle b, 1 \rangle$ for some + $b \in D$ with $b < 1$. Now $\bigvee A = \langle 1, 1\rangle$ + in $D^{2}$ and the dual argument shows that $\bigwedge A$ also + exists in $D^{2}$. Hence $D$ is complete. Conditions~(1) and + (2) are obvious for $D^{\langle 2 \rangle}$. +\end{proof} + +\begin{corollary}\label{C:prime} + If $D$ is complete-prime, then so is $D^{\langle 2 \rangle}$. +\end{corollary} + +The motivation for the following result comes from Soo-Key Foo~\cite{sF90}. + +\begin{lemma}\label{L:ccr} + Let $\Theta$ be a complete congruence relation of + $D^{\langle 2 \rangle}$ such that + \begin{equation}\label{E:rigid} + \langle 1, d \rangle \equiv \langle 1, 1 \rangle \pmod{\Theta}, + \end{equation} + for some $d \in D$ with $d < 1$. Then $\Theta = \iota$. +\end{lemma} + +\begin{proof} + Let $\Theta$ be a complete congruence relation of + $D^{\langle 2 \rangle}$ satisfying \eqref{E:rigid}. Then $\Theta = +\iota$. +\end{proof} + +\section{The $\Pi^{*}$ construction}\label{S:P*} +The following construction is crucial to our proof of the Main Theorem: + +\begin{definition}\label{D:P*} + Let $D_{i}$, for $i \in I$, be complete distributive lattices + satisfying condition~\textup{(2)}. Their $\Pi^{*}$ product is defined +as + follows: + \[ + \Pi^{*} ( D_{i} \mid i \in I ) = \Pi ( D_{i}^{-} \mid i \in I ) + 1; + \] + that is, $\Pi^{*} ( D_{i} \mid i \in I )$ is $\Pi ( D_{i}^{-} \mid + i \in I )$ with a new unit element. +\end{definition} + +\begin{notation} + If $i \in I$ and $d \in D_{i}^{-}$, then + \[ + \langle \dots, 0, \dots, \overset{i}{d}, \dots, 0, \dots \rangle + \] + is the element of $\Pi^{*} ( D_{i} \mid i \in I )$ whose $i$-th + component is $d$ and all the other components are $0$. +\end{notation} + +See also Ernest~T. Moynahan \cite{eM57a}. Next we verify: + +\begin{theorem}\label{T:P*} + Let $D_{i}$, for $i \in I$, be complete distributive lattices + satisfying condition~\textup{(2)}. Let $\Theta$ be a complete +congruence + relation on $\Pi^{*} ( D_{i} \mid i \in I )$. If there exist + $i \in I$ and $d \in D_{i}$ with $d < 1_{i}$ such that for + all $d \leq c < 1_{i}$, + \begin{equation}\label{E:cong1} + \langle \dots, 0, \dots,\overset{i}{d}, + \dots, 0, \dots \rangle \equiv \langle \dots, 0, \dots, + \overset{i}{c}, \dots, 0, \dots \rangle \pmod{\Theta}, + \end{equation} + then $\Theta = \iota$. +\end{theorem} + +\begin{proof} + Since + \begin{equation}\label{E:cong2} + \langle \dots, 0, \dots, \overset{i}{d}, \dots, 0, + \dots \rangle \equiv \langle \dots, 0, \dots, + \overset{i}{c}, \dots, 0, \dots \rangle \pmod{\Theta}, + \end{equation} + and $\Theta$ is a complete congruence relation, it follows from + condition~(3) that + \begin{align}\label{E:cong} + & \langle \dots, \overset{i}{d}, \dots, 0, + \dots \rangle \equiv\\ + &\qquad \quad \bigvee ( \langle \dots, 0, \dots, + \overset{i}{c}, \dots, 0, \dots \rangle \mid d \leq c < 1 ) + \equiv 1 \pmod{\Theta}. \notag + \end{align} + + Let $j \in I$ for $j \neq i$, and let $a \in D_{j}^{-}$. + Meeting both sides of the congruence \eqref{E:cong2} with + $\langle \dots, 0, \dots, \overset{j}{a}, \dots, 0, \dots \rangle$, + we obtain + \begin{align}\label{E:comp} + 0 &= \langle \dots, 0, \dots, \overset{i}{d}, \dots, 0, \dots + \rangle \wedge \langle \dots, 0, \dots, \overset{j}{a}, \dots, 0, + \dots \rangle\\ + &\equiv \langle \dots, 0, \dots, \overset{j}{a}, \dots, 0, \dots + \rangle \pmod{\Theta}. \notag + \end{align} + Using the completeness of $\Theta$ and \eqref{E:comp}, we get: + \[ + 0 \equiv \bigvee ( \langle \dots, 0, \dots, \overset{j}{a}, + \dots, 0, \dots \rangle \mid a \in D_{j}^{-} ) = 1 \pmod{\Theta}, + \] + hence $\Theta = \iota$. +\end{proof} + +\begin{theorem}\label{T:P*a} + Let $D_{i}$ for $i \in I$ be complete distributive lattices + satisfying conditions \textup{(2)} and \textup{(3)}. Then + $\Pi^{*} ( D_{i} \mid i \in I )$ also satisfies conditions \textup{(2)} + and \textup{(3)}. +\end{theorem} + +\begin{proof} + Let $\Theta$ be a complete congruence on + $\Pi^{*} ( D_{i} \mid i \in I )$. Let $i \in I$. Define + \[ + \widehat{D}_{i} = \{ \langle \dots, 0, \dots, \overset{i}{d}, + \dots, 0, \dots \rangle \mid d \in D_{i}^{-} \} \cup \{ 1 \}. + \] + Then $\widehat{D}_{i}$ is a complete sublattice of + $\Pi^{*} ( D_{i} \mid i \in I )$, and $\widehat{D}_{i}$ is + isomorphic to $D_{i}$. Let $\Theta_{i}$ be the restriction of + $\Theta$ to $\widehat{D}_{i}$. + + Since $D_{i}\) is complete-simple, so is $\widehat{D}_{i}$, and + hence $\Theta_{i}$ is $\omega$ or $\iota$. If + $\Theta_{i} = \rho$ for all $i \in I$, then + $\Theta = \omega$. If there is an $i \in I$, such that + $\Theta_{i} = \iota$, then $0 \equiv 1 \pmod{\Theta}$, hence + $\Theta = \iota$. +\end{proof} + +The Main Theorem follows easily from Theorems~\ref{T:P*} and \ref{T:P*a}. + +\begin{thebibliography}{9} + + \bibitem{sF90} + Soo-Key Foo, \emph{Lattice Constructions,} Ph.D. thesis, University + of Winnebago, Winnebago, MN, December, 1990. + + \bibitem{gM68} + George~A. Menuhin, \emph{Universal Algebra,} D.~van Nostrand, + Princeton-Toronto-London-Mel\-bourne, 1968. + + \bibitem{eM57} + Ernest~T. Moynahan, \emph{On a problem of M.H. Stone,} Acta Math. + Acad.Sci. Hungar. \textbf{8} (1957), 455--460. + + \bibitem{eM57a} + \bysame, \emph{Ideals and congruence relations in lattices.~II,} + Magyar Tud. Akad. Mat. Fiz. Oszt. K\"{o}zl. \textbf{9} (1957), + 417--434 (Hungarian). + + \bibitem{fR82} + Ferenc~R. Richardson, \emph{General Lattice Theory,} Mir, Moscow, + expanded and revised ed., 1982 (Russian). + +\end{thebibliography} +\end{document} + diff --git a/info/examples/FirstSteps/sampart2.tex b/info/examples/FirstSteps/sampart2.tex new file mode 100644 index 0000000000..27a24e3a88 --- /dev/null +++ b/info/examples/FirstSteps/sampart2.tex @@ -0,0 +1,252 @@ +% Sample file: sampart2.tex +% The sample article for the amsart document class +% with user-defined commands +% Typeset with LaTeX format + +\documentclass{amsart} +\usepackage{amssymb,latexsym} +\usepackage{lattice} + +\theoremstyle{plain} +\newtheorem{theorem}{Theorem} +\newtheorem{corollary}{Corollary} +\newtheorem*{main}{Main~Theorem} +\newtheorem{lemma}{Lemma} +\newtheorem{proposition}{Proposition} + +\theoremstyle{definition} +\newtheorem{definition}{Definition} + +\theoremstyle{remark} +\newtheorem*{notation}{Notation} + +\numberwithin{equation}{section} + +\newcommand{\Prodm}[2]{\gP(\,#1\mid#2\,)} + % product with a middle +\newcommand{\Prodsm}[2]{\gP^{*}(\,#1\mid#2\,)} + % product * with a middle +\newcommand{\vct}[2]{\vv<\dots,0,\dots,\overset{#1}{#2},% +\dots,0,\dots>}% special vector +\newcommand{\fp}{\F{p}}% Fraktur p +\newcommand{\Ds}{D^{\langle2\rangle}} + +\begin{document} +\title[Complete-simple distributive lattices] + {A construction of complete-simple\\ + distributive lattices} +\author{George~A. Menuhin} +\address{Computer Science Department\\ + University of Winnebago\\ + Winnebago, Minnesota 23714} +\email{menuhin@ccw.uwinnebago.edu} +\urladdr{http://math.uwinnebago.ca/homepages/menuhin/} +\thanks{Research supported by the NSF under grant number~23466.} +\keywords{Complete lattice, distributive lattice, complete + congruence, congruence lattice} +\subjclass{Primary: 06B10; Secondary: 06D05} +\date{March 15, 1995} + +\begin{abstract} + In this note we prove that there exist \emph{complete-simple + distributive lattices,} that is, complete distributive + lattices in which there are only two complete congruences. +\end{abstract} +\maketitle + +\section{Introduction}\label{S:intro} +In this note we prove the following result: + +\begin{main} + There exists an infinite complete distributive lattice + $K$ with only the two trivial complete congruence relations. +\end{main} + +\section{The $\Ds$ construction}\label{S:Ds} +For the basic notation in lattice theory and universal algebra, +see Ferenc~R. Richardson~\cite{fR82} and George~A. Menuhin~\cite{gM68}. +We start with some definitions: + +\begin{definition}\label{D:prime} + Let $V$ be a complete lattice, and let $\fp = [u, v]$ be + an interval of $V$. Then $\fp$ is called + \emph{complete-prime} if the following three conditions are satisfied: + \begin{enumerate} + \item[(1)] $u$ is meet-irreducible but $u$ is \emph{not} + completely meet-irreducible; + \item[(2)] $v$ is join-irreducible but $v$ is \emph{not} + completely join-irreducible; + \item[(3)] $[u, v]$ is a complete-simple lattice. + \end{enumerate} +\end{definition} + +Now we prove the following result: + +\begin{lemma}\label{L:ds} + Let $D$ be a complete distributive lattice satisfying + conditions~\textup{(1)} and~\textup{(2)}. + Then $\Ds$ is a sublattice of $D^{2}$; hence $\Ds$ is + a lattice, and $\Ds$ is a complete distributive lattice + satisfying conditions~~\textup{(1)} and~~\textup{(2)}. +\end{lemma} + +\begin{proof} + By conditions~(1) and (2), $\Ds$ is a sublattice of + $D^{2}$. Hence, $\Ds$ is a lattice. + + Since $\Ds$ is a sublattice of a distributive lattice, $\Ds$ is + a distributive lattice. Using the characterization of + standard ideals in Ernest~T. Moynahan~\cite{eM57}, + $\Ds$ has a zero and a unit element, namely, + $\vv<0, 0>$ and $\vv<1, 1>$. To show that $\Ds$ is + complete, let $\es \ne A \ci \Ds$, and let $a = \JJ A$ + in $D^{2}$. If $a \in \Ds$, then + $a = \JJ A$ in $\Ds$; otherwise, $a$ is of the form + $\vv$ for some $b \in D$ with $b < 1$. Now + $\JJ A = \vv<1, 1>$ in $D^{2}$, and + the dual argument shows that $\MM A$ also exists in + $D^{2}$. Hence $D$ is complete. Conditions~(1) and (2) + are obvious for $\Ds$. +\end{proof} + +\begin{corollary}\label{C:prime} + If $D$ is complete-prime, then so is $\Ds$. +\end{corollary} + +The motivation for the following result comes from Soo-Key +Foo~\cite{sF90}. + +\begin{lemma}\label{L:ccr} + Let $\gQ$ be a complete congruence relation of $\Ds$ such + that + \begin{equation}\label{E:rigid} + \vv<1, d> \equiv \vv<1, 1> \pod{\gQ}, + \end{equation} + for some $d \in D$ with $d < 1$. Then $\gQ = \gi$. +\end{lemma} + +\begin{proof} + Let $\gQ$ be a complete congruence relation of $\Ds$ + satisfying \eqref{E:rigid}. Then $\gQ = \gi$. +\end{proof} + +\section{The $\gP^{*}$ construction}\label{S:P*} +The following construction is crucial to our proof of the +Main~Theorem: + +\begin{definition}\label{D:P*} + Let $D_{i}$, for $i \in I$, be complete distributive + lattices satisfying condition~\tup{(2)}. Their $\gP^{*}$ + product is defined as follows: + \[ + \Prodsm{ D_{i} }{i \in I} = \Prodm{ D_{i}^{-} }{i \in I} +1; + \] + that is, $\Prodsm{ D_{i} }{i \in I}$ is + $\Prodm{ D_{i}^{-} }{i \in I}$ with a new unit element. +\end{definition} + +\begin{notation} + If $i \in I$ and $d \in D_{i}^{-}$, then + \[ + \vct{i}{d} + \] + is the element of $\Prodsm{ D_{i} }{i \in I}$ whose + $i$-th component is $d$ and all the other + components are $0$. +\end{notation} + +See also Ernest~T. Moynahan~\cite{eM57a}. Next we verify: + +\begin{theorem}\label{T:P*} + Let $D_{i}$, for $i \in I$, be complete distributive + lattices satisfying condition~\tup{(2)}. Let $\gQ$ be a + complete congruence relation on + $\Prodsm{ D_{i} }{i \in I}$. If there exist + $i \in I$ and $d \in D_{i}$ with $d < 1_{i}$ such + that for all $d \leq c < 1_{i}$, + \begin{equation}\label{E:cong1} + \vct{i}{d} \equiv \vct{i}{c} \pod{\gQ}, + \end{equation} + then $\gQ = \gi$. +\end{theorem} + +\begin{proof} + Since + \begin{equation}\label{E:cong2} + \vct{i}{d} \equiv \vct{i}{c} \pod{\gQ}, + \end{equation} + and $\gQ$ is a complete congruence relation, it follows + from condition~(3) that + \begin{align}\label{E:cong} + &\vct{i}{d} \equiv \notag\\ + &\qq\q{\JJm{\vct{i}{c}}{d \leq c < 1}=1} \pod{\gQ}. + \end{align} + Let $j \in I$ for $j \neq i$, and let + $a \in D_{j}^{-}\). Meeting both sides of the congruence + \eqref{E:cong} with $\vct{j}{a}$, we obtain + \begin{align}\label{E:comp} + 0 &= \vct{i}{d} \mm \vct{j}{a}\\ + &\equiv \vct{j}{a}\pod{\gQ}. \notag + \end{align} + Using the completeness of $\gQ$ and \eqref{E:comp}, we get: + \begin{equation}\label{E:cong3} + 0=\JJm{ \vct{j}{a} }{ a \in D_{j}^{-} } \equiv 1 \pod{\gQ}, + \end{equation} + hence $\gQ = \gi$. +\end{proof} + +\begin{theorem}\label{T:P*a} + Let $D_{i}$ for $i \in I$ be complete distributive + lattices satisfying + conditions~\tup{(2)} and \tup{(3)}. Then + $\Prodsm{ D_{i} }{i \in I}$ also satisfies + conditions~\tup{(2)} and \tup{(3)}. +\end{theorem} + +\begin{proof} + Let $\gQ$ be a complete congruence on + $\Prodsm{ D_{i} }{i \in I}$. Let $i \in I$. Define + \begin{equation}\label{E:dihat} + \widehat{D}_{i} = \setm{ \vct{i}{d} }{ d \in D_{i}^{-} } + \uu \set{1}. + \end{equation} + Then $\widehat{D}_{i}$ is a complete sublattice of + $\Prodsm{ D_{i} }{i \in I}$, and $\widehat{D}_{i}$ + is isomorphic to $D_{i}$. Let $\gQ_{i}$ be the + restriction of $\gQ$ to $\widehat{D}_{i}$. Since + $D_{i}$ is complete-simple, so is $\widehat{D}_{i}$, + hence $\gQ_{i}$ is $\go$ or $\gi$. If $\gQ_{i} = \go$ + for all $i \in I$, then $\gQ = \go$. + If there is an $i \in I$, such that $\gQ_{i} = \gi$, + then $0 \equiv 1 \pod{\gQ}$, and hence $\gQ = \gi$. +\end{proof} + +The Main Theorem follows easily from Theorems~\ref{T:P*} and +\ref{T:P*a}. + +\begin{thebibliography}{9} + + \bibitem{sF90} + Soo-Key Foo, \emph{Lattice Constructions,} Ph.D. thesis, University + of Winnebago, Winnebago, MN, December, 1990. + + \bibitem{gM68} + George~A. Menuhin, \emph{Universal Algebra,} D.~van Nostrand, + Princeton-Toronto-London-Mel\-bourne, 1968. + + \bibitem{eM57} + Ernest~T. Moynahan, \emph{On a problem of M.H. Stone,} Acta Math. + Acad.Sci. Hungar. \textbf{8} (1957), 455--460. + + \bibitem{eM57a} + \bysame, \emph{Ideals and congruence relations in lattices.~II,} + Magyar Tud. Akad. Mat. Fiz. Oszt. K\"{o}zl. \textbf{9} (1957), + 417--434 (Hungarian). + + \bibitem{fR82} + Ferenc~R. Richardson, \emph{General Lattice Theory,} Mir, Moscow, + expanded and revised ed., 1982 (Russian). + +\end{thebibliography} + +\end{document} \ No newline at end of file -- cgit v1.2.3