mjo-algebra.tex

   1 %
   2 % Abstract algebraic structures.
   3 %
   4 \ifx\havemjoalgebra\undefined
   5 \def\havemjoalgebra{1}
   6
   7
   8 \ifx\operatorname\undefined
   9   \usepackage{amsopn}
  10 \fi
  11
  12 \input{mjo-common} % for \of, and \binopmany
  13
  14
  15 % The additive identity element of its argument, which should be
  16 % an algebraic structure.
  17 \newcommand*{\zero}[1]{ 0_{{#1}} }
  18
  19 \ifdefined\newglossaryentry
  20   \newglossaryentry{zero}{
  21     name={\ensuremath{\zero{R}}},
  22     description={the additive identity element of $R$},
  23     sort=z
  24   }
  25 \fi
  26
  27 % The multiplicative identity element of its argument, which should be
  28 % an algebraic structure.
  29 \newcommand*{\unit}[1]{ 1_{{#1}} }
  30
  31 \ifdefined\newglossaryentry
  32   \newglossaryentry{unit}{
  33     name={\ensuremath{\unit{R}}},
  34     description={the multiplicative identity (unit) element of $R$},
  35     sort=u
  36   }
  37 \fi
  38
  39 % The direct sum of two things.
  40 \newcommand*{\directsum}[2]{ {#1}\oplus{#2} }
  41
  42 % The direct sum of three things.
  43 \newcommand*{\directsumthree}[3]{ \directsum{#1}{\directsum{#2}{#3}} }
  44
  45 % The (indexed) direct sum of many things.
  46 \newcommand*{\directsummany}[3]{ \binopmany{\bigoplus}{#1}{#2}{#3} }
  47
  48
  49 % The (sub)algebra generated by its argument, a subset of some ambient
  50 % algebra. By definition this is the smallest subalgebra (of the
  51 % ambient one) containing that set.
  52 \newcommand*{\alg}[1]{\operatorname{alg}\of{{#1}}}
  53 \ifdefined\newglossaryentry
  54   \newglossaryentry{alg}{
  55     name={\ensuremath{\alg{X}}},
  56     description={the (sub)algebra generated by $X$},
  57     sort=a
  58   }
  59 \fi
  60
  61
  62 % The fraction field of its argument, an integral domain. The name
  63 % "Frac" was chosen here instead of "Quot" because the latter
  64 % corresponds to the term "quotient field," which can be mistaken in
  65 % some cases for... a quotient field (something mod something).
  66 \newcommand*{\Frac}[1]{\operatorname{Frac}\of{{#1}}}
  67
  68 % The ideal generated by its argument, a subset consisting of ring or
  69 % algebra elements.
  70 \newcommand*{\ideal}[1]{\operatorname{ideal}\of{{#1}}}
  71 \ifdefined\newglossaryentry
  72   \newglossaryentry{ideal}{
  73     name={\ensuremath{\ideal{X}}},
  74     description={the ideal generated by $X$},
  75     sort=i
  76   }
  77 \fi
  78
  79
  80 % The polynomial ring whose underlying commutative ring of
  81 % coefficients is the first argument and whose indeterminates (a
  82 % comma-separated list) are the second argumnt.
  83 \newcommand*{\polyring}[2]{{#1}\left[{#2}\right]}
  84 \ifdefined\newglossaryentry
  85   \newglossaryentry{polyring}{
  86     name={\ensuremath{\polyring{R}{X}}},
  87     description={polynomials with coefficients in $R$ and variable $X$},
  88     sort=p
  89   }
  90 \fi
  91
  92
  93 % The stabilizer subgroup of its first argument that fixes the point
  94 % given by its second argument.
  95 \newcommand*{\Stab}[2]{ #1_{#2} }
  96
  97
  98 % The affine algebraic variety consisting of the common solutions to
  99 % every polynomial in its argument, which should be a subset of some
 100 % polynomial ring.
 101 \newcommand*{\variety}[1]{ \mathcal{V}\of{{#1}} }
 102 \ifdefined\newglossaryentry
 103   \newglossaryentry{variety}{
 104     name={\ensuremath{\variety{I}}},
 105     description={variety corresponding to the ideal $I$},
 106     sort=v
 107   }
 108 \fi
 109
 110 \fi