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\section*{modular lattice}

\begin{quote}%
This is about a notion in \emph{[[order theory]]/[[logic]]}. For an unrelated notion of a similar name in [[group theory]]/[[quadratic form]]-theory see at \emph{[[modular integral lattice]]}.


\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{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{alternative_formulations}{Alternative formulations}\dotfill \pageref*{alternative_formulations} \linebreak
\noindent\hyperlink{proofs_of_equivalence}{Proofs of equivalence}\dotfill \pageref*{proofs_of_equivalence} \linebreak
\noindent\hyperlink{derivation_of_modular_identity}{Derivation of modular identity}\dotfill \pageref*{derivation_of_modular_identity} \linebreak
\noindent\hyperlink{modular_law__modular_identity}{Modular law $\Leftrightarrow$ modular identity}\dotfill \pageref*{modular_law__modular_identity} \linebreak
\noindent\hyperlink{freyds_modular_law__modular_identity}{Freyd's modular law $\Leftrightarrow$ modular identity}\dotfill \pageref*{freyds_modular_law__modular_identity} \linebreak
\noindent\hyperlink{modular_identity__definition_1}{Modular identity $\Rightarrow$ definition 1}\dotfill \pageref*{modular_identity__definition_1} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{characterization}{Characterization}\dotfill \pageref*{characterization} \linebreak
\noindent\hyperlink{free_modular_lattices}{Free modular lattices}\dotfill \pageref*{free_modular_lattices} \linebreak
\noindent\hyperlink{see_also}{See also}\dotfill \pageref*{see_also} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{modular lattice} is a lattice where ``opposite sides'' of a ``diamond'' formed by four points $x \wedge y$, $x$, $y$, $x \vee y$ are ``congruent''.

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

A modular lattice is a [[lattice]] which satisfies a modular law, which we introduce after a few preliminaries.

In any lattice $L$, given two elements $x, y \in L$ with $x \leq y$, let $[x, y]$ denote the [[interval]] $\{z \colon x \leq z \leq y\}$. Then, given any two elements $a, b \in L$, there is an [[adjunction|adjoint pair]]

\begin{displaymath}
a \vee (-) \colon [a \wedge b, b] \leftrightarrows [a, a \vee b] \colon (-) \wedge b
\end{displaymath}
where $a \vee (-)$ is left adjoint to $(-) \wedge b$. Indeed, for any $w \in [a \wedge b, b]$, we have a [[adjunction|unit]]

\begin{displaymath}
w \leq (a \vee w) \wedge b,
\end{displaymath}
whereas for any $z \in [a, a \vee b]$, we have dually a [[adjunction|counit]]

\begin{displaymath}
a \vee (z \wedge b) \leq z.
\end{displaymath}
\begin{defn}
\label{}\hypertarget{}{}
A lattice $L$ is \textbf{modular} if for any $a, b \in L$, the adjoint pair

\begin{displaymath}
a \vee (-) \dashv (-) \wedge b \colon [a, a \vee b] \to [a \wedge b, b]
\end{displaymath}
is an [[adjoint equivalence]].

\end{defn}
This is perhaps the most memorable definition for a category theorist: it is a precise expression of the slogan given in the Idea section.

It is immediate that the concept of modular lattice is \textbf{[[duality|self-dual]]}, i.e., if $L$ is modular, then so is $L^{op}$.

\hypertarget{alternative_formulations}{}\subsection*{{Alternative formulations}}\label{alternative_formulations}

In the lattice-theoretic literature, modularity is usually formulated somewhat differently. Here are three alternative conditions on a lattice, all equivalent to that of Definition 1.

\begin{enumerate}%
\item The \textbf{modular law} is the universal [[Horn theory|Horn sentence]]

\begin{displaymath}
a \leq b \vdash (a \vee z) \wedge b = a \vee (z \wedge b).
\end{displaymath}

\item The \textbf{modular identity} is the universal equation

\begin{displaymath}
(a \vee z) \wedge (a \vee b) = a \vee (z \wedge (a \vee b))
\end{displaymath}

\item ``\textbf{Freyd's modular law}'' (for lack of better term; see [[allegory]]) is the universal inequality

\begin{displaymath}
(a \vee z) \wedge b \leq a \vee (z \wedge (a \vee b)).
\end{displaymath}


\end{enumerate}
\hypertarget{proofs_of_equivalence}{}\subsubsection*{{Proofs of equivalence}}\label{proofs_of_equivalence}

\hypertarget{derivation_of_modular_identity}{}\paragraph*{{Derivation of modular identity}}\label{derivation_of_modular_identity}

To see that the modular identity follows from Definition 1, observe that for any $z \in L$ we have

\begin{displaymath}
a \leq (a \vee z) \wedge (a \vee b) \leq a \vee b
\end{displaymath}
Let $w = (a \vee z) \wedge (a \vee b)$. Under $(-) \wedge b \colon [a, a \vee b] \to [a \wedge b, b]$, this element $w$ is sent to

\begin{displaymath}
(a \vee z) \wedge (a \vee b) \wedge b = (a \vee z) \wedge b.
\end{displaymath}
Under Definition 1, this last element is sent back to $w$ by $a \vee (-)$. Therefore we have

\begin{displaymath}
(a \vee z) \wedge (a \vee b) = w = a \vee ((a \vee z) \wedge b)
\end{displaymath}
and since this is true for all $a, b, z$, we can interchange $z$ and $b$ and rearrange by commutativity to get

\begin{displaymath}
(a \vee z) \wedge (a \wedge b) = a \vee (z \wedge (a \vee b))
\end{displaymath}
which is the modular identity.

\hypertarget{modular_law__modular_identity}{}\paragraph*{{Modular law $\Leftrightarrow$ modular identity}}\label{modular_law__modular_identity}

To get the modular law from the modular identity, just use the fact that the hypothesis $a \leq b$ is equivalent to $a \vee b = b$, and use this to substitute $b$ for $a \vee b$ in the modular identity. Conversely, from the tautology $a \leq a \vee b$, we can instantiate the modular law to derive the modular identity.

\hypertarget{freyds_modular_law__modular_identity}{}\paragraph*{{Freyd's modular law $\Leftrightarrow$ modular identity}}\label{freyds_modular_law__modular_identity}

From the tautology $(a \vee z) \wedge b \leq (a \vee z) \wedge (a \vee b)$, it is clear that Freyd's modular law follows from the modular identity. Conversely, by substituting $a \vee b$ for $b$ in Freyd's modular law, we derive the special case

\begin{displaymath}
(a \vee z) \wedge (a \vee b) \leq a \vee (z \wedge (a \vee b))
\end{displaymath}
whereas the opposite inequality

\begin{displaymath}
a \vee (z \wedge (a \vee b)) \leq (a \vee z) \wedge (a \vee b)
\end{displaymath}
holds in any lattice, so the modular identity follows from Freyd's modular law.

\hypertarget{modular_identity__definition_1}{}\paragraph*{{Modular identity $\Rightarrow$ definition 1}}\label{modular_identity__definition_1}

Finally, we derive the adjoint equivalence of Definition 1 from the modular identity. One half of the adjoint equivalence states that if $a \leq z \leq a \vee b$, then $z = a \vee (z \wedge b)$; if this holds, then the other half follows because it is the dual statement. If $a \leq z \leq a \vee b$, then

\begin{displaymath}
z = (a \vee b) \wedge z = (a \vee b) \wedge (a \vee z)
\end{displaymath}
just by the laws of a lattice. By the modular identity (again switching $b$ and $z$), the right side equals $a \vee (b \wedge (a \vee z))$. But since $a \vee z = z$, this equals $a \vee (b \wedge z) = a \vee (z \wedge b)$, as was to be shown.

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

\begin{itemize}%
\item Every [[distributive lattice]], e.g., a [[Heyting algebra]], is modular. Indeed, if $a \leq b$ in a distributive lattice, we have

\begin{displaymath}
(a \vee z) \wedge b = (a \wedge b) \vee (z \wedge b) = a \vee (z \wedge b)
\end{displaymath}
which proves the modular law.


\item For any [[Mal'cev variety]] or Mal'cev [[algebraic theory]], the lattice of internal [[equivalence relations]] of an algebra is a modular lattice. The equivalence classes often arise as cosets of kernels; for example, for a [[vector space]] $V$, equivalence relations correspond to subspaces of $V$, and form a modular lattice. Other examples include the lattice of [[normal subgroups]] of a [[group]], the lattice of two-sided [[ideals]] of a [[ring]], etc.


\item In fact, any lattice of commuting equivalence relations on a [[set]] is a modular lattice (being a suballegory of the [[allegory]] of sets, one in which composition provides the join).


\item Every abstract [[projective plane]] gives rise to a modular lattice $L$ whose underlying set is the disjoint union

\begin{displaymath}
\{0\} \cup \{1\} \cup \{points\} \cup \{lines\}
\end{displaymath}
where $0$ is taken as bottom, $1$ as top, the points are atoms, and the lines are coatoms, ordered by the incidence relation. The projective plane need not be Desarguesian.


\item [[Young–Fibonacci lattice]]



\end{itemize}
\hypertarget{characterization}{}\subsection*{{Characterization}}\label{characterization}

The smallest non-modular lattice has 5 elements and is called the \emph{pentagon}, denoted $N_5$. It can be described as the lattice $\{\bot, a, b, c, \top\}$ where $b \leq c$ and $a$ is incomparable with $b$ and $c$.

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Dedekind)} A lattice $L$ is modular if and only if there is no injective function $f \colon N_5 \to L$ that preserves meets and joins.

\end{theorem}
(Notice we are leaving out the condition of preservation of the top and bottom elements.) The direction $\Rightarrow$ is easy enough: $L$ being modular is incompatible with an injective $f: N_5 \to L$ preserving meets and joins, since we contradict the modular law in $L$ by applying $f$ to $(b \vee a) \wedge c = \top \wedge c = c$ and $b \vee (a \wedge c) = b \vee \bot = b$.

This is reminiscent of forbidden minor characterizations of certain classes of graphs; see [[graph minor]]. There is a similar ``forbidden sublattice'' characterization of [[distributive lattices]] -- see this \hyperlink{Lein}{comment} by Tom Leinster at the $n$-Category Caf\'e{}.

\hypertarget{free_modular_lattices}{}\subsection*{{Free modular lattices}}\label{free_modular_lattices}

Free modular lattices tend to be complicated. \hyperlink{Ded}{Dedekind} showed that the free modular lattice on 3 elements has 28 elements; its [[Hasse diagram]] can be seen in these \href{http://www.math.hawaii.edu/~jb/math618/os9uh.pdf}{lecture notes} by J.B. Nation (chapter 9, page 100).

N.B.: this notion of lattice is meant with respect to the signature $(\wedge, \vee)$; if we include top and bottom constants in the signature, then the free modular lattice on three elements has 30 elements. A compelling illustration (in gif form) which exhibits [[triality]] of this lattice is given in this $n$-Category Caf\'e{} \href{https://golem.ph.utexas.edu/category/2015/09/the_free_modular_lattice_on_3.html#c049649}{post}, as part of a larger discussion which explores the connection with linear representations of the [[quiver]] $D_4$ (the [[Coxeter diagram]] of $SO(8)$).

For $n \geq 4$, the free modular lattice generated by $n$ elements is infinite and in fact has an undecidable word problem (Freese, Herrmann).

\hypertarget{see_also}{}\subsection*{{See also}}\label{see_also}

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


\item [[orthomodular lattice]]


\item [[distributive lattice]]


\item [[geometric lattice]]



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

\begin{itemize}%
\item [[Richard Dedekind]], ``\"U{}ber die von drei Moduln erzeugte Dualgruppe gemeinsamen Teiler'', Math. Annalen 53 (1900), 371--403, reprinted in \emph{Gesammelte mathematische Werke}, Vol. 2, pp. 236--271, Chelsea, New York, 1968.

\end{itemize}
\begin{itemize}%
\item C. Herrmann, ``On the word problem for the modular lattice with four free generators'', Mathematische Annalen 265 (1983), 513-527. (\href{http://www.springerlink.com/content/w862594668m2n375/}{Springerlink})


\item J.B. Nation, \emph{Revised Notes on Lattice Theory}. Available here: (\href{http://www.math.hawaii.edu/~jb/}{web})


\item [[Tom Leinster]], Comment on Sol\`e{}r's Theorem, December 4, 2010. (\href{http://golem.ph.utexas.edu/category/2010/12/solers_theorem.html#c035900}{link})



\end{itemize}
[[!redirects modular lattices]]



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