Order, lattices, ordered algebraic structures Modular and complemented lattices Example

Pentagon lattice N5

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. For instance $a \vee b = c \vee b = 1$ and $a \wedge b = c \wedge b = 0$ .

$N_5$ is not a modular lattice. 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, 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.

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Lattice concept
Modular lattice concept
Pentagon lattice N5 example

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Tag	`DHC8TZXGYS`
URI	`m2://DHC8TZXGYS`
Kind	example
Writing	W0 stub
MSC	06C05, 06B05
Instance of	Lattice
Not an instance of	Modular lattice (note: with a below c, the modular law fails for x = b.)
Uses	Lattice, Modular lattice

Pentagon lattice N5

Example

Remarks

Reading path

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