From 17fd11a2ac39c8d20680a61279efed7bd18f93f2 Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Fri, 1 Nov 2019 17:14:15 -0400
Subject: [PATCH] mjo-algebra: add \alg{} for the subalgebra generated by a
set.
---
examples.tex | 8 ++++++--
mjo-algebra.tex | 5 +++++
2 files changed, 11 insertions(+), 2 deletions(-)
diff --git a/examples.tex b/examples.tex
index af125a7..c00279b 100644
--- a/examples.tex
+++ b/examples.tex
@@ -30,8 +30,12 @@
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 $\Frac{R}$. 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.
+ 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}
diff --git a/mjo-algebra.tex b/mjo-algebra.tex
index 4f3f79c..71ba622 100644
--- a/mjo-algebra.tex
+++ b/mjo-algebra.tex
@@ -9,6 +9,11 @@
\usepackage{amsopn}
\fi
+% The (sub)algebra generated by its argument, a subset of some ambient
+% algebra. By definition this is the smallest subalgebra (of the
+% ambient one) containing that set.
+\newcommand*{\alg}[1]{\operatorname{alg}\of{{#1}}}
+
% The fraction field of its argument, an integral domain. The name
% "Frac" was chosen here instead of "Quot" because the latter
% corresponds to the term "quotient field," which can be mistaken in
--
2.45.3