X-Git-Url: http://gitweb.michael.orlitzky.com/?p=mjotex.git;a=blobdiff_plain;f=mjo-algebra.tex;h=25b96d9f8a184edc051ed2867bc8fa9b4b0a269b;hp=4f3f79cd5dab03694d66be516bb4b157c7c1e1c9;hb=HEAD;hpb=42bef52e8272b97d0fae6243dd7621b793086e28

diff --git a/mjo-algebra.tex b/mjo-algebra.tex
index 4f3f79c..3bb51c1 100644
--- a/mjo-algebra.tex
+++ b/mjo-algebra.tex
@@ -9,6 +9,56 @@
   \usepackage{amsopn}
 \fi
 
+\input{mjo-common} % for \of, and \binopmany
+
+
+% The additive identity element of its argument, which should be
+% an algebraic structure.
+\newcommand*{\zero}[1]{ 0_{{#1}} }
+
+\ifdefined\newglossaryentry
+  \newglossaryentry{zero}{
+    name={\ensuremath{\zero{R}}},
+    description={the additive identity element of $R$},
+    sort=z
+  }
+\fi
+
+% 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} }
+
+% The direct sum of three things.
+\newcommand*{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
+
+% The (indexed) direct sum of many things.
+\newcommand*{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
+
+
+% 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}}}
+\ifdefined\newglossaryentry
+  \newglossaryentry{alg}{
+    name={\ensuremath{\alg{X}}},
+    description={the (sub)algebra generated by $X$},
+    sort=a
+  }
+\fi
+
+
 % 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
@@ -18,11 +68,43 @@
 % The ideal generated by its argument, a subset consisting of ring or
 % algebra elements.
 \newcommand*{\ideal}[1]{\operatorname{ideal}\of{{#1}}}
+\ifdefined\newglossaryentry
+  \newglossaryentry{ideal}{
+    name={\ensuremath{\ideal{X}}},
+    description={the ideal generated by $X$},
+    sort=i
+  }
+\fi
+
 
 % The polynomial ring whose underlying commutative ring of
 % coefficients is the first argument and whose indeterminates (a
 % comma-separated list) are the second argumnt.
 \newcommand*{\polyring}[2]{{#1}\left[{#2}\right]}
+\ifdefined\newglossaryentry
+  \newglossaryentry{polyring}{
+    name={\ensuremath{\polyring{R}{X}}},
+    description={polynomials with coefficients in $R$ and variable $X$},
+    sort=p
+  }
+\fi
+
+
+% The stabilizer subgroup of its first argument that fixes the point
+% given by its second argument.
+\newcommand*{\Stab}[2]{ #1_{#2} }
 
 
+% The affine algebraic variety consisting of the common solutions to
+% every polynomial in its argument, which should be a subset of some
+% polynomial ring.
+\newcommand*{\variety}[1]{ \mathcal{V}\of{{#1}} }
+\ifdefined\newglossaryentry
+  \newglossaryentry{variety}{
+    name={\ensuremath{\variety{I}}},
+    description={variety corresponding to the ideal $I$},
+    sort=v
+  }
+\fi
+
 \fi