design/tpl: Emphasize commutativity of monoids in classification system

design/tpl/sec/class.tex

@@ -96,8 +96,8 @@ $\alpha$~serves as a placeholder for an optional \xml{any="true"},
 Note the wildcard variable matching \xmlattr{desc}---%
   its purpose is only to provide documentation.
 
-\begin{corollary}[$\odot$ Monoid]\corlabel{odot-monoid}
-  $\odot$ is a monoid in \axmref{class-intro}.
+\begin{corollary}[$\odot$ Commutative Monoid]\corlabel{odot-monoid}
+  $\odot$ is a commutative monoid in \axmref{class-intro}.
 \end{corollary}
 \begin{proof}
   By \axmref{class-intro},
@@ -112,6 +112,15 @@ Note the wildcard variable matching \xmlattr{desc}---%
       which is proved by \lemref{monoid-land}.
 \end{proof}
 
+\index{classification!commutativity}
+\index{compiler!classification commutativity}
+While \axmref{class-intro} seems to imply an ordering to matches,
+  users of the language are free to specify matches in any order
+    and the compiler will rearrange matches as it sees fit.
+This is due to the commutativity of~$\odot$ as proved by
+  \corref{odot-monoid},
+    and not only affords great ease of use to users of~\tame{},
+      but also great flexibility to compiler writers.
 
 
 \def\cpredmatseq{{M^0_j}_k \bullet\cdots\bullet {M^l_j}_k}

design/tpl/sec/notation.tex

@@ -339,6 +339,12 @@ Given that, we have $f\bicomp{[]} = f\bicomp{[A]}$ for functions returning
 \index{abstract algebra!semigroup}
 Monoids originate from abstract algebra.
 A monoid is a semigroup with an added identity element~$e$.
+Only the identity element must be commutative,
+  but if the binary operation~$\bullet$ is \emph{also} commutative,
+    then the monoid is a \dfn{commutative monoid}.\footnote{%
+      A commutative monoid is less frequently referred to as an
+        \dfn{abelian monoid},
+          related to the common term \dfn{abelian group}.}
 
 Consider some sequence of operations
   $x_0 \bullet\cdots\bullet x_n \in S$.
@@ -384,21 +390,21 @@ If $x=\Set{}$,
 
 \index{conjunction!monoid}
 \begin{lemma}\lemlabel{monoid-land}
-  $\Monoid\Bool\land\true$ is a monoid.
+  $\Monoid\Bool\land\true$ is a commutative monoid.
 \end{lemma}
 \begin{proof}
-  $\Monoid\Bool\land\true$ is associative by \dfnref{conj-assoc}.
-  The identity element is~$\true\in\Bool$ by \dfnref{conj-commut} and
-    \dfnref{conj-simpl},
-      as in $\true \land p \equiv p \land \true \equiv p$.
+  $\Monoid\Bool\land\true$ is associative by \dfnref{conj-assoc}
+    and commutative by \dfnref{conj-commut}.
+  The identity element is~$\true\in\Bool$ by \dfnref{conj-simpl}.
 \end{proof}
 
 \index{disjunction!monoid}
 \begin{lemma}\lemlabel{monoid-lor}
-  $\Monoid\Bool\lor\false$ is a monoid.
+  $\Monoid\Bool\lor\false$ is a commutative monoid.
 \end{lemma}
 \begin{proof}
-  $\Monoid\Bool\lor\false$ is associative by \dfnref{disj-assoc}.
+  $\Monoid\Bool\lor\false$ is associative by \dfnref{disj-assoc}
+    and commutative by \dfnref{disj-commut}.
   The identity $\false\in\Bool$ follows from
 
   \begin{alignat*}{3}

design/tpl/tpl.sty

@@ -140,6 +140,9 @@
 
 \let\xml\texttt
 
+% Definitions (introduction of terms)
+\let\dfn\textsl
+
 % Symbols appear at the beginning of the index
 \newcommand\indexsym[2]{\index{__sym_#2@{\ensuremath{#1}}|see {#2}}}