mjo-algebra: add \alg{} for the subalgebra generated by a set.

[mjotex.git] / examples.tex

diff --git a/examples.tex b/examples.tex
index 4005cdd984dfe9a46194751b97b1bea2b8679748..c00279b16beb44514ae8e80e371d8dc3f8392a35 100644 (file)
--- a/examples.tex
+++ b/examples.tex
@@ -1,7 +1,17 @@
 \documentclass{report}
 
-% We have to load this before mjotex so that mjotex knows to define
-% its glossary entries.
+% Setting hypertexnames=false forces hyperref to use a consistent
+% internal counter for proposition/equation references rather than
+% being clever, which doesn't work after we reset those counters.
+\usepackage[hypertexnames=false]{hyperref}
+\hypersetup{
+  colorlinks=true,
+  linkcolor=blue,
+  citecolor=blue
+}
+
+% We have to load this after hyperref, so that links work, but before
+% mjotex so that mjotex knows to define its glossary entries.
 \usepackage[nonumberlist]{glossaries}
 \makenoidxglossaries
 
@@ -19,7 +29,13 @@
     If $R$ is a \index{commutative ring}, then $\polyring{R}{X,Y,Z}$
     is a multivariate polynomial ring with indeterminates $X$, $Y$,
     and $Z$, and coefficients in $R$. If $R$ is a moreover an integral
-    domain, then its fraction field is $\Frac{R}$.
+    domain, then its fraction field is $\Frac{R}$. If $x,y,z \in R$,
+    then $\ideal{\set{x,y,z}}$ is the ideal generated by
+    $\set{x,y,z}$, which is defined to be the smallest ideal in $R$
+    containing that set. Likewise, if we are in an algebra
+    $\mathcal{A}$ and if $x,y,z \in \mathcal{A}$, then
+    $\alg{\set{x,y,z}}$ is the smallest subalgebra of $\mathcal{A}$
+    containing the set $\set{x,y,z}$.
   \end{section}
 
   \begin{section}{Algorithm}
@@ -43,8 +59,9 @@
   \end{section}
 
   \begin{section}{Arrow}
-    The identity operator on $V$ is $\identity{V}$. The composition of
-    $f$ and $g$ is $\compose{f}{g}$. The inverse of $f$ is
+    The constant function that always returns $a$ is $\const{a}$. The
+    identity operator on $V$ is $\identity{V}$. The composition of $f$
+    and $g$ is $\compose{f}{g}$. The inverse of $f$ is
     $\inverse{f}$. If $f$ is a function and $A$ is a subset of its
     domain, then the preimage under $f$ of $A$ is $\preimage{f}{A}$.
   \end{section}
@@ -57,9 +74,31 @@
   \begin{section}{Common}
     The function $f$ applied to $x$ is $f\of{x}$. We can group terms
     like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
-    set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. Here's a pair
-    of things $\pair{1}{2}$ or a triple of them $\triple{1}{2}{3}$,
-    and the factorial of the number $10$ is $\factorial{10}$.
+    set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. The tuples go
+    up to seven, for now:
+    %
+    \begin{itemize}
+      \begin{item}
+        Pair: $\pair{1}{2}$,
+      \end{item}
+      \begin{item}
+        Triple: $\triple{1}{2}{3}$,
+      \end{item}
+      \begin{item}
+        Quadruple: $\quadruple{1}{2}{3}{4}$,
+      \end{item}
+      \begin{item}
+        Qintuple: $\quintuple{1}{2}{3}{4}{5}$,
+      \end{item}
+      \begin{item}
+        Sextuple: $\sextuple{1}{2}{3}{4}{5}{6}$,
+      \end{item}
+      \begin{item}
+        Septuple: $\septuple{1}{2}{3}{4}{5}{6}{7}$.
+      \end{item}
+    \end{itemize}
+    %
+    The factorial of the number $10$ is $\factorial{10}$.
 
     The Cartesian product of two sets $A$ and $B$ is
     $\cartprod{A}{B}$; if we take the product with $C$ as well, then
@@ -157,6 +196,9 @@
     $\boundedops[W]{V}$. If $W = V$, then we write $\boundedops{V}$
     instead.
 
+    If you want to solve a system of equations, try Cramer's
+    rule~\cite{ehrenborg}.
+
     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
     but what if $W = V^{\perp}$? Then we wish to indicate that fact by
     writing $\directsumperp{V}{W}$. That operator should survive a
@@ -335,5 +377,8 @@
   \glsaddall
   \printnoidxglossaries
 
+  \bibliographystyle{mjo}
+  \bibliography{local-references}
+
   \printindex
 \end{document}