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\begin{document}

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\section*{lattice}

\begin{quote}%
This entry is about the notion in \emph{[[order theory]]/[[logic]]}. For other notions of the same name, such as in [[bilinear form]]-theory, see at \emph{[[lattice (disambiguation)]]}.


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{bounded_lattices_and_pseudolattices}{Bounded lattices and pseudolattices}\dotfill \pageref*{bounded_lattices_and_pseudolattices} \linebreak
\noindent\hyperlink{lattice_homomorphisms}{Lattice homomorphisms}\dotfill \pageref*{lattice_homomorphisms} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{lattice} is a [[partial order|poset]] which admits all finite [[meets]] and finite [[joins]] (or all finite [[products]] and finite [[coproducts]], regarding a poset as a [[category]] (a [[(0,1)-category]])).

A \textbf{lattice} can also be defined as an algebraic structure, with the binary operations $\wedge$ and $\vee$ and the constants $\top$ and $\bot$. (These correspond, respectively, to binary and nullary meets and joins in the poset-theoretic definition; accordingly, they are read `meet', `join', `[[top]]', and `[[bottom]]'.) Here are the axioms for these operations:

\begin{itemize}%
\item $\wedge$ and $\vee$ are each [[idempotent]], [[commutative magma|commutative]], and [[associative magma|associative]];
\item the \emph{absorption laws}: $a \vee (a \wedge b) = a$, and $a \wedge (a \vee b) = a$;
\item $\top$ and $\bot$ are the respective [[identity element|identities]] of $\wedge$ and $\vee$.

\end{itemize}
You can recover the original poset from either the meet or the join; $a \leq b$ iff $a \wedge b = a$, and $b \leq a$ iff $a \vee b = a$, and then prove that $a\wedge b$ is the greatest lower bound for $a$, $b$ and $a \vee b$ is the least upper bound for $a$, $b$. (Notice that the absorption laws guarantee that these two descriptions of $\leq$ agree.) Indeed, we may say that a lattice is a \emph{bisemilattice} in that it has two semilattice structures that are compatible in that they define (but in dual ways) the same partial order.

Note that a poset with only finite meets \emph{or} finite joins is a (meet- or join-) [[semilattice]], while a lattice which has \emph{all} joins and meets (not just finitary ones) is a [[complete lattice]].

\hypertarget{bounded_lattices_and_pseudolattices}{}\subsection*{{Bounded lattices and pseudolattices}}\label{bounded_lattices_and_pseudolattices}

Traditionally, a lattice need have only finite [[inhabited set|inhabited]] meets and joins; that is, it need not have a top or bottom element. Algebraically, this means $\wedge$ and $\vee$ need not have identities.

Then one may call a lattice that \emph{does} have a top and a bottom a \textbf{bounded lattice}; in general, a [[bounded poset]] is a poset that has top and bottom elements.

The other approach is to define a lattice, as above, to require a top and a bottom and then use the term \textbf{pseudolattice} to allow for the possibility that it might not.

From an algebraic point of view, requiring top and bottom is quite natural, a special case of preferring [[monoids]] to more general [[semigroups]]. In any case, one can formally adjoin a top and a bottom if required. On the other hand, many examples, especially from analysis, do not come with a top or a bottom, and adjoining them would break the other structure. For example, adjoining top ($\infty$) and bottom ($-\infty$) to the [[real line]] makes it no longer a [[field]] (addition is especially problematic); more generally, a [[Banach lattice]] need not (and, except in one degenerate case, cannot) have a top or a bottom.

\hypertarget{lattice_homomorphisms}{}\subsection*{{Lattice homomorphisms}}\label{lattice_homomorphisms}

A lattice homomorphism $f$ from a lattice $A$ to a lattice $B$ is a [[function]] from $A$ to $B$ (seen as sets) that preserves $\wedge$ and $\vee$ (and $\top$ and $\bot$, if these are required):

\begin{displaymath}
f(x \wedge y) = f(x) \wedge f(y),\; f(\top) = \top,\; f(x \vee y) = f(x) \vee f(y),\; f(\bot) = \bot .
\end{displaymath}
Note that such a homomorphism is necessarily a [[monotone function]], but the converse fails.

Thus, a lattice is a poset (or even a semilattice) with [[property-like structure]].

Lattices and lattice homomorphims form a [[concrete category]] [[Lat]].

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

\begin{itemize}%
\item [[complete lattice]]


\item [[distributive lattice]]


\item [[modular lattice]]


\item [[orthomodular lattice]]


\item [[geometric lattice]]


\item [[continuous lattice]]


\item [[complemented lattice]]


\item [[semilattice]]


\item [[suplattice]]


\item [[Hilbert lattice]]


\item [[frame]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Peter Johnstone]], chapter 1 of \emph{[[Stone Spaces]]}


\item [[Jacob Lurie]], section A.1.1 of \emph{[[Spectral Algebraic Geometry]]}



\end{itemize}
[[!redirects lattices]] [[!redirects bisemilattice]] [[!redirects bounded lattice]] [[!redirects pseudolattice]]



\end{document}