From f82e8e51ac99547bab1a02d678a1476971f17444 Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Tue, 10 Dec 2019 15:28:47 -0500
Subject: [PATCH] Add \unit{} for the multiplicative identity element.

Technically we could pass off \identity{X} as the multiplicative
identity element in an algebraic structure, by using the "arrow"
interpretation in the right category. But let's not try to shoot
ourselves in the foot too hard, yeah?
---
 examples.tex    |  3 +++
 mjo-algebra.tex | 12 ++++++++++++
 mjo-arrow.tex   | 10 +++++++++-
 3 files changed, 24 insertions(+), 1 deletion(-)

diff --git a/examples.tex b/examples.tex
index 131a213..65e0c80 100644
--- a/examples.tex
+++ b/examples.tex
@@ -36,6 +36,9 @@
     $\mathcal{A}$ and if $x,y,z \in \mathcal{A}$, then
     $\alg{\set{x,y,z}}$ is the smallest subalgebra of $\mathcal{A}$
     containing the set $\set{x,y,z}$.
+
+    If $R$ has a multiplicative identity (that is, a unit) element,
+    then that element is denoted by $\unit{R}$.
   \end{section}
 
   \begin{section}{Algorithm}
diff --git a/mjo-algebra.tex b/mjo-algebra.tex
index 7ff7b5a..36ad4bd 100644
--- a/mjo-algebra.tex
+++ b/mjo-algebra.tex
@@ -12,6 +12,18 @@
 \input{mjo-common} % for \of, and \binopmany
 
 
+% The multiplicative identity element of its argument, which should be
+% an algebraic structure.
+\newcommand*{\unit}[1]{ 1_{{#1}} }
+
+\ifdefined\newglossaryentry
+  \newglossaryentry{unit}{
+    name={\ensuremath{\unit{R}}},
+    description={the multiplicative identity (unit) element of $R$},
+    sort=u
+  }
+\fi
+
 % The direct sum of two things.
 \newcommand*{\directsum}[2]{ {#1}\oplus{#2} }
 
diff --git a/mjo-arrow.tex b/mjo-arrow.tex
index 9bf9dc5..2d5095e 100644
--- a/mjo-arrow.tex
+++ b/mjo-arrow.tex
@@ -23,9 +23,17 @@
   }
 \fi
 
-% The identity operator/arrow on its argument.
+% The identity function/arrow on its argument.
 \newcommand*{\identity}[1]{ \operatorname{id}_{{#1}} }
 
+\ifdefined\newglossaryentry
+  \newglossaryentry{identity}{
+    name={\ensuremath{\identity{X}}},
+    description={the identity function or arrow on $X$},
+    sort=i
+  }
+\fi
+
 % The composition of two arrows/functions. For example, the
 % composition of g with f is \compose{g}{f}\of{x} === g\of{f\of{x}}.
 \newcommand*{\compose}[2]{ {#1}\circ{#2} }
-- 
2.43.2