other.tex

   1 \section{áÌÇÅÂÒÁÉÞÅÓËÉÅ ÓÔÒÕËÔÕÒÙ}
   2 \label{sec:alg-structures}
   3
   4 ðÒÅÄÕÐÒÅÖÄÅÎÉÅ: ÜÔÏ ÎÅ ÔÏÞÎÙÅ ÏÐÒÅÄÅÌÅÎÉÑ ÁÌÇÅÂÒÁÉÞÅÓËÉÈ ÏÂßÅËÔÏ×,
   5 ÓËÏÒÅÅ, ÜÔÏ~--- ÎÁÐÏÍÉÎÁÎÉÅ ÏÓÎÏ×ÎÙÈ ÉÄÅÊ ÏÐÒÅÄÅÌÅÎÉÊ.
   6
   7 \subsection{çÒÕÐÐÙ É~Ô.Ð.}
   8
   9 \newcommand*{\setBool}[2]{%
  10   \ifthenelse{#2}%
  11   {\setboolean{#1}{true}}%
  12   {\setboolean{#1}{false}}}
  13 \newcommand*{\initBool}[2]{\newboolean{#1}\setBool{#1}{#2}}
  14 \newcommand*{\gAxiom}[2]{{\small{#1}:} {#2}}
  15 \newcommand*{\Group}[4]{{%
  16     \initBool{gA}{\NOT\equal{#1}{}}%
  17     \newcommand*{\gOp}{#1}%
  18     \initBool{gE}{\NOT\equal{#2}{}}%
  19     \newcommand*{\gId}{#2}%
  20     \initBool{gI}{\NOT\equal{#3{}}{{}}}%
  21     \newcommand*{\gInv}[1]{#3{##1}}%
  22     \initBool{gC}{\equal{#4}{C}}%
  23     \begin{enumerate}
  24     \item\ifthenelse{\boolean{gA}}{
  25         \gAxiom{ÁÓÓÏÃ-ÎÏÓÔØ}% ÁÓÓÏÃÉÁÔÉ×ÎÏÓÔØ
  26         {$x \gOp (y \gOp z) = (x \gOp y) \gOp z$}
  27       }{}
  28       \ifthenelse{\boolean{gE}\OR\boolean{gI}\OR\boolean{gC}}{
  29     \item\ifthenelse{\boolean{gE}}{
  30         \gAxiom{ÎÅÊÔÒ.\ ÐÏ~$\gOp$}%
  31         {$x \gOp \gId = \gId \gOp x = x$}
  32       }{}
  33       \ifthenelse{\boolean{gI}\OR\boolean{gC}}{
  34       \item\ifthenelse{\boolean{gI}}{
  35           \gAxiom{ÏÂÒÁÔ.\ ÐÏ~$\gOp$}%
  36           {$\gInv{x} \gOp x = x \gOp \gInv{x} = \gId$}
  37         }{}
  38         \ifthenelse{\boolean{gC}}{
  39           \item \gAxiom{ËÏÍÍÕÔ-ÎÏÓÔØ}%
  40             {$x \gOp y = y \gOp x$}
  41         }{}
  42       }{}
  43     }{}
  44   \end{enumerate}
  45 }}
  46 \newcommand*{\Ring}[9]{{%
  47   \initBool{rD}{\equal{#9}{D}}%
  48   \begin{tabular}{|p{.45\textwidth}|p{.45\textwidth}|}
  49     \hline
  50     \textbf{ÐÏ~$#1$:} & \textbf{ÐÏ~$#5$ (ÂÅÚ ÜÌÅÍÅÎÔÁ $#2$):}\\
  51     %\hline
  52     \Group{#1}{#2}{#3}{#4}
  53     &
  54     \Group{#5}{#6}{#7}{#8}
  55     \ifthenelse{\boolean{rD}}{
  56       \\
  57     %\hline
  58       \multicolumn{2}{|l|}{\gAxiom{ÄÉÓÔÒÉÂÕÔÉ×ÎÏÓÔØ}%
  59         {$(x #1 y) #5 z = (x #5 z) #1 (x #5 z)$,
  60           $z #5 (x #1 y) = (z #5 x) #1 (z #5 y)$}}\\
  61     }{}
  62     \hline
  63   \end{tabular}
  64 }}
  65
  66 \newcommand*{\inv}[1]{#1^{-1}}
  67 \newcommand*{\opp}[1]{(-#1)}
  68
  69 \begin{description}
  70 \item[\tING{ÇÒÕÐÐÁ}] $\Stru{G; \circ, e, \inv{\place} }$:
  71 \Group{\circ}{e}{\inv}{nC}
  72
  73 \item[\tING{ËÏÍÍÕÔÁÔÉ×ÎÁÑ  ÇÒÕÐÐÁ}], ÉÌÉ \tING{ÁÂÅÌÅ×Á},
  74 $\Stru{G; \circ, e, \inv{\place} }$:
  75 \Group{\circ}{e}{\inv}{C}
  76
  77 \item[\tING{ÐÏÌÕÇÒÕÐÐÁ}]
  78 $\Stru{G; \circ }$:
  79 \Group{\circ}{}{}{nC}
  80
  81 \item[\tING{ÐÏÌÕÇÒÕÐÐÁ Ó ÅÄÉÎÉÃÅÊ}], ÉÌÉ
  82 \tING{ÍÏÎÏÉÄ},
  83 $\Stru{ G; \circ, e }$:
  84 \Group{\circ}{e}{}{nC}
  85
  86 \item[\tING{ËÏÍÍÕÔÁÔÉ×ÎÙÊ ÍÏÎÏÉÄ}]
  87 $\Stru{ G; \circ, e}$:
  88 \Group{\circ}{e}{}{C}
  89 \end{description}
  90
  91 \paragraph{ðÒÉÍÅÒÙ ÍÏÎÏÉÄÏ×.}
  92
  93 \todo{[ðÒÉÍÅÒÙ ÉÚ~\cite[Lecture~2]{KA-lect02}]}
  94
  95 \begin{example}
  96   \begin{itemize}
  97   \item
  98     ÷ÓÅ ÉÚ×ÅÓÔÎÙÅ ÇÒÕÐÐÙ.
  99   \item
 100   ó×ÏÂÏÄÎÙÊ ÍÏÎÏÉÄ Ó ÏÐÅÒÁÃÉÅÊ ËÏÎËÁÔÅÎÁÃÉÉ,
 101   ÐÏÒÏÖÄ£ÎÎÙÊ $\Sigma = \{ a_1,
 102   \dotsc, a_r \}$ (<<ÍÎÏÖÅÓÔ×Ï ÓÌÏ× × ÁÌÆÁ×ÉÔÅ $\Sigma$>>):
 103   $\Stru{ \Sigma^*, \cdot, \epsilon }$.
 104   \end{itemize}
 105 \end{example}
 106
 107 \paragraph{ðÒÉÍÅÒÙ ËÏÍÍÕÔÁÔÉ×ÎÙÈ ÍÏÎÏÉÄÏ×}
 108
 109 \begin{example}
 110   \begin{itemize}
 111   \item
 112     ÷ÓÅ ÉÚ×ÅÓÔÎÙÅ ËÏÍÍÕÔÁÔÉ×ÎÙÅ ÇÒÕÐÐÙ.
 113   \item
 114     ó×ÏÂÏÄÎÙÊ ËÏÍÍÕÔÁÔÉ×ÎÙÊ ÍÏÎÏÉÄ Ó ÏÐÅÒÁÃÉÅÊ ËÏÎËÁÔÅÎÁÃÉÉ,
 115     ÐÏÒÏÖÄ£ÎÎÙÊ $\Sigma = \{ a_1,
 116     \dotsc, a_r \}$: $\Stru{ \Sigma^\oplus, \cdot, \epsilon } $
 117     (ÉÚÏÍÏÒÆÅÎ $\Stru{ (\mathbb N \cup \{ 0 \})^r, +, \mathbf 0 }$).
 118   \end{itemize}
 119 \end{example}
 120
 121 \subsubsection{ëÏÌØÃÁ É~Ô.Ð.}
 122
 123 %% \begin{description}
 124 %% \item[\tING{ËÏÌØÃÏ}]
 125 %% $\Stru{ K; +, 0, \opp{\place}; \cdot, 1 }$:\\
 126 %% \Ring{+}{0}{\opp}{C}{\cdot}{1}{}{nC}{D}
 127
 128 %% \item[\tING{ÐÏÌÕËÏÌØÃÏ}]
 129 %% $\Stru{ K; +, 0; \cdot, 1 }$:\\
 130 %% \Ring{+}{0}{}{C}{\cdot}{1}{}{nC}{D}
 131
 132 %% \item[\tING{ÐÏÌÅ}]
 133 %% $\Stru{ F; +, 0, \opp{\place}; \cdot, 1, \inv{\place} }$:\\
 134 %% \Ring{+}{0}{\opp}{C}{\cdot}{1}{\inv}{C}{D}
 135
 136 %% \item[\tING{ËÏÌØÃÏ Ó ÄÅÌÅÎÉÅÍ}] (\tING{division ring})
 137 %% $\Stru{ F; +, 0, \opp{\place}; \cdot, 1, \inv{\place} }$:\\
 138 %% \Ring{+}{0}{\opp}{C}{\cdot}{1}{\inv}{nC}{D}
 139 %% \end{description}
 140
 141 \begin{definition}\label{def:ISr}
 142 ëÏÒÏÔËÏ, \tING{ÉÄÅÍÐÏÔÅÎÔÎÏÅ ÐÏÌÕËÏÌØÃÏ}\T ÜÔÏ ÓÔÒÕËÔÕÒÁ $(S; +,
 143 \cdot, 0, 1)$, ÔÁËÁÑ, ÞÔÏ
 144 $\Stru{S; +, 0}$ Ñ×ÌÑÅÔÓÑ \tND{×ÅÒÈÎÅÊ
 145   ÐÏÌÕÒÅÛ£ÔËÏÊ}{def:upper-semilattice} Ó ÎÁÉÍÅÎØÛÉÍ ÜÌÅÍÅÎÔÏÍ~0,
 146 $\Stru{S; \cdot, 1}$\T \tND{ÍÏÎÏÉÄÏÍ}{def:monoid},
 147 0 Ñ×ÌÑÅÔÓÑ ÁÎÎÉÇÉÌÑÔÏÒÏÍ ÄÌÑ $\cdot$,
 148 É $\cdot$ ÄÉÓÔÒÉÂÕÔÉ×ÎÁ ÏÔÎÏÓÉÔÅÌØÎÏ $+$ ÓÌÅ×Á É
 149 ÓÐÒÁ×Á. \cite{KA-modular-elimination}
 150 \end{definition}
 151
 152 óÔÒÕËÔÕÒÁ ×ÅÒÈÎÅÊ ÐÏÌÕÒÅÛ£ÔËÉ~$\Stru{S; +, 0}$ ÉÎÄÕÃÉÒÕÅÔ
 153 ÎÁ ÌÀÂÏÍ ÉÄÅÍÐÏÔÅÎÔÎÏÍ ÐÏÌÕËÏÌØÃÅ
 154 ÅÓÔÅÓÔ×ÅÎÎÙÊ ÞÁÓÔÉÞÎÙÊ ÐÏÒÑÄÏË:
 155 \begin{equation}
 156   \label{eq:ISr-order}
 157   x \leq y \equivdef x + y = y.
 158 \end{equation}
 159
 160 þÁÓÔÏ ×ÍÅÓÔÏ $x \cdot y$ ÐÉÛÕÔ $xy$.
 161
 162
 163 %% \section{áÌÇÅÂÒÙ ëÌÉÎÉ}
 164 %% \label{sec:KA-more}
 165
 166
 167 %% äÒÕÇÉÅ ÐÒÉÍÅÒÙ
 168
 169
 170
 171 %%% Local Variables:
 172 %%% mode: latex
 173 %%% TeX-master: "main"
 174 %%% End: