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\begin{document}

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\section*{meet}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - contents]]

\hypertarget{category_theory}{}\paragraph*{{$(0,1)$-Category theory}}\label{category_theory}

[[!include (0,1)-category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{infimum_of_real_numbers}{Infimum of real numbers}\dotfill \pageref*{infimum_of_real_numbers} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In a [[preordered set]] or [[partially ordered set]] then the \emph{meet} or \emph{infimum} of a [[subset]] of elements is, if it exists, the largest element in the set which is smaller or equal to all the elements in the subset. If this element is member of the original subset, then it is also called the \emph{[[minimum]]} of that subset.

If we think of the pre-ordered set as a [[category]] (a [[(0,1)-category]]) then the meet is the [[limit]] over the given subset, if it exists, regarded as a [[diagram]]. Thus in a [[partially ordered set]] this is unique if it exists, otherwise it is unique up to [[isomorphism]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

If $x$ and $y$ are elements of a [[partial order|poset]], then their \textbf{meet}, or \textbf{infimum}, is an element $x \wedge y$ of the poset such that:

\begin{itemize}%
\item $x \wedge y \leq x$ and $x \wedge y \leq y$;
\item if $a \leq x$ and $a \leq y$, then $a \leq x \wedge y$. Such a meet may not exist; if it does, then it is unique.

\end{itemize}
In a [[preorder|proset]], a meet may be defined similarly, but it need not be unique. (However, it is still unique up to the natural [[equivalence]] in the proset.)

The above definition is for the meet of two elements of a poset, but it can easily be generalised to any number of elements. It may be more common to use `meet' for a meet of finitely many elements and `infimum' for a meet of (possibly) infinitely many elements, but they are the same concept. The meet may also be called the \textbf{minimum} if it equals one of the original elements.

A poset that has all finite meets is a \textbf{meet-[[semilattice]]}. A poset that has all infima is an \textbf{[[inflattice]]}.

A meet of [[subsets]] or [[subobjects]] is called an [[intersection]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item A meet of no elements is a [[top]] element.


\item Any element $a$ is a meet of that one element.



\end{itemize}
\hypertarget{infimum_of_real_numbers}{}\subsubsection*{{Infimum of real numbers}}\label{infimum_of_real_numbers}

Often one considers infima of subsets of the [[real numbers]] $\mathbb{R}$, regarded with their canonical [[preordering]], which in this case is in fact a [[total order]].

For $S \subset \mathbb{R}$ a subset, say that a \emph{lower bound} is an element $b \in \mathbb{R}$ such that $\underset{s \in S \subset \mathbb{R}}{\forall}( b \leq s )$.

Then the infimum of $S$ is, if it exists, that lower bound $inf(S)$ of $S$ such that for $b$ any other lower bound of $S$ then $b \leq inf(S)$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

As a poset is a special kind of [[category]], a meet is simply a [[product]] in that category.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[join]]


\item \textbf{meet}



\end{itemize}
[[!redirects meets]] [[!redirects infimum]] [[!redirects infimums]] [[!redirects infima]] [[!redirects inf]] [[!redirects infs]]



\end{document}