---
tag: DHC8TZXGYS
kind: example
name: Pentagon lattice N5
msc: [06C05, 06B05]
instance_of: [WH4SFAACAS]
not_instance_of:
  - target: QX7WZ92QAK
    note: with a below c, the modular law fails for x = b
uses: [WH4SFAACAS, QX7WZ92QAK]
sources:
  - url: https://en.wikipedia.org/wiki/Modular_lattice
---

## Example

The *pentagon lattice* $N_5$ has five elements $0, a, b, c, 1$.
Its order is generated by $0 < a < c < 1$ and $0 < b < 1$, and $b$ is incomparable to $a$ and to $c$.
Every pair of elements has a least upper bound and a greatest lower bound, so $N_5$ is a [[lattice|WH4SFAACAS]].
For instance $a \vee b = c \vee b = 1$ and $a \wedge b = c \wedge b = 0$.

$N_5$ is not a [[modular lattice|QX7WZ92QAK]].
Take $a \le c$ and $x = b$.
Then $a \vee (b \wedge c) = a \vee 0 = a$, while $(a \vee b) \wedge c = 1 \wedge c = c \ne a$.

## Remarks

By [[Dedekind's characterization|ANRXAG227H]], a lattice is modular exactly when it has no sublattice isomorphic to $N_5$.
Since every distributive lattice is modular, $N_5$ is also not distributive.
The Hasse diagram of $N_5$ is a pentagon, which gives the lattice its name.

## Explanations

### L1 human

Draw a pentagon: a bottom, a top, a short side with one middle point $b$ and a long side with two points $a$ below $c$.
Going up from $a$ through $b$ reaches the top, and coming back down to the long side lands at $c$, not at $a$.
This mismatch is exactly the failure of the modular law.
It is the smallest lattice where this happens.