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\import{order/order.tex}
\import{function.tex}
\begin{struct}\label{meet_semilattice}
A meet semilattice $X$ is an ordered set
equipped with
\begin{enumerate}
\item $\meet$
\end{enumerate}
such that
\begin{enumerate}
\item\label{meet_type} for all $x,y\in \carrier[X]$ we have
$\meet[X](x,y)\in \carrier[X]$.
\item\label{meet_lb} for all $x,y\in \carrier[X]$ we have
$\meet[X](x,y) \mathrel{\lt[X]} x, y$.
\item\label{meet_glb} for all $a,x,y\in \carrier[X]$ such that $a\mathrel{\lt[X]} x, y$ we have
$a\mathrel{\lt[X]} \meet[X](x,y)$.
\end{enumerate}
\end{struct}
\begin{proposition}\label{meet_idempotent}
Let $X$ be a meet semilattice.
Let $x\in\carrier[X]$.
Then $\meet[X](x,x) = x$.
\end{proposition}
\begin{proof}
We have $\meet[X](x,x)\in\carrier[X]$ by \cref{meet_type}.
We have $\meet[X](x,x)\mathrel{\lt[X]}x$ by \cref{meet_lb}.
We have $x\mathrel{\lt[X]}x$
by \cref{meet_semilattice,orderedset,quasiorder_refl,reflexive_on}.
Thus $x\mathrel{\lt[X]}\meet[X](x,x)$ by \cref{meet_glb}.
Follows by \cref{meet_semilattice,orderedset,orderedset_antisym,antisymmetric}.
\end{proof}
%\begin{proposition}\label{meet_comm}
% Let $X$ be a meet semilattice.
% Suppose $x, y\in X$.
% Then $\meet(x,y) = \meet(y,x)$.
%\end{proposition}
%\begin{proposition}\label{meet_assoc}
% Let $X$ be a meet semilattice.
% Suppose $x, y, z\in X$.
% Then $\meet(x,\meet(y,z)) = \meet(\meet(x,y),z)$.
%\end{proposition}
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