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diff --git a/library/test.tex b/library/test.tex deleted file mode 100644 index d30bbba..0000000 --- a/library/test.tex +++ /dev/null @@ -1,54 +0,0 @@ -\import{algebra/semigroup.tex} -\section{monoid} - -\begin{struct}\label{monoid} - A monoid $A$ is a semigroup equipped with - \begin{enumerate} - \item $\neutral$ - \end{enumerate} - such that - \begin{enumerate} %muss hier ein enumerate hin - \item\label{monoid_type} $\neutral[A]\in \carrier[A]$. - \item\label{monoid_right} for all $a\in \carrier[A]$ we have $\mul[A](a,\neutral[A]) = a$. - \item\label{monoid_left} for all $a\in \carrier[A]$ we have $\mul[A](\neutral[A], a) = a$. - \end{enumerate} -\end{struct} - - -\section{Group} - -\begin{struct}\label{group} - A group $A$ is a monoid such that - \begin{enumerate} - \item\label{group_inverse} for all $a \in \carrier[A]$ there exist $b \in \carrier[A]$ such that $\mul[A](a, b) =\neutral[A]$. - \end{enumerate} -\end{struct} - -\begin{abbreviation}\label{cfourdot} - $a\cdot b = \mul(a,b)$. -\end{abbreviation} - -\begin{lemma}\label{neutral_is_idempotent} - Let $G$ be a group. $\neutral[G]$ is a idempotent element of $G$. -\end{lemma} - -\begin{proposition}\label{leftinverse_eq_rightinverse} - Let $G$ be a group and assume $a \in G$. - Then there exist $b\in G$ - such that $a \cdot b = \neutral[G]$ and $b \cdot a = \neutral[G]$. -\end{proposition} -\begin{proof} - There exist $b \in G$ such that $a \cdot b = \neutral[G]$. - There exist $c \in G$ such that $b \cdot c = \neutral[G]$. - $a \cdot b = \neutral[G]$. - $(a \cdot b) \cdot c = (\neutral[G]) \cdot c$. - $(a \cdot b) \cdot c = a \cdot (b \cdot c)$. - $a \cdot \neutral[G] = \neutral[G] \cdot c$. - $c = c \cdot \neutral[G]$. - $c = \neutral[G] \cdot c$. - $a \cdot \neutral[G] = c \cdot \neutral[G]$. - $a \cdot \neutral[G] = c$ by \cref{monoid_right}. - $a = c$ by \cref{monoid_right}. - $b \cdot a = b \cdot c$. - $b \cdot a = \neutral[G]$. -\end{proof}
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