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diff --git a/library/algebra/monoid.tex b/library/algebra/monoid.tex new file mode 100644 index 0000000..06fcb50 --- /dev/null +++ b/library/algebra/monoid.tex @@ -0,0 +1,19 @@ +\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} + \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} + +\begin{corollary}\label{monoid_implies_semigroup} + Let $A$ be a monoid. Then $A$ is a semigroup. +\end{corollary}
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