design/tpl (Monoids and Sequences): Add missing index entries

Forgot in previous commit.
master
Mike Gerwitz 2021-05-14 10:38:17 -04:00
parent 8d54420656
commit 9fd57872ed
1 changed files with 8 additions and 0 deletions

View File

@ -323,6 +323,8 @@ Given that, we have $f\bicomp{[]} = f\bicomp{[A]}$ for functions returning
\subsection{Monoids and Sequences}
\index{abstract algebra!monoid}
\index{monoid|see abstract algebra, monoid}
\begin{definition}[Monoid]\dfnlabel{monoid}
Let $S$ be some set. A \emph{monoid} is a triple $\Monoid S\bullet e$
with the axioms
@ -337,6 +339,8 @@ Given that, we have $f\bicomp{[]} = f\bicomp{[A]}$ for functions returning
\end{align}
\end{definition}
\index{abstract algebra}
\index{abstract algebra!semigroup}
Monoids originate from abstract algebra.
A monoid is a semigroup with an added identity element~$e$.
@ -350,6 +354,8 @@ When the sequence has one or zero elements,
as $x_0 \bullet e = x_0$ in the case of one element
or $e \bullet e = e$ in the case of zero.
\indexsym\cdots{sequence}
\index{sequence}
Generally,
given some monoid $\Monoid S\bullet e$ and a sequence $\Fam{x}jJ\in S$
where $n<|J|$,
@ -378,6 +384,7 @@ If $x=\Set{1}$,
If $x=\Set{}$,
we have $0$.
\index{conjunction!monoid}
\begin{lemma}
$\Monoid\Bool\land\true$ is a monoid.
\end{lemma}
@ -388,6 +395,7 @@ If $x=\Set{}$,
as in $\true \land p \equiv p \land \true \equiv p$.
\end{proof}
\index{disjunction!monoid}
\begin{lemma}
$\Monoid\Bool\lor\false$ is a monoid.
\end{lemma}