# isDistributive -- determines if a lattice is distributive

## Synopsis

• Usage:
i = isDistributive P
• Inputs:
• P, an instance of the type Poset, a lattice
• Outputs:
• i, , whether $P$ is distributive

## Description

The lattice $P$ is distributive if the meet operation distributes over the join operation. Equivalently, $P$ is distributive if the join operation distributes over the meet operation.

The $n$ booleanLattice is distributive.

 i1 : isDistributive booleanLattice 3 o1 = true

The pentagon lattice and diamond lattice are prototypical non-distributive lattices.

 i2 : P = poset {{1,2}, {1,3}, {3,4}, {2,5}, {4, 5}}; i3 : isLattice P o3 = true i4 : isDistributive P o4 = false i5 : D = poset {{1,2}, {1,3}, {1,4}, {2,5}, {3,5}, {4,5}}; i6 : isLattice D o6 = true i7 : isDistributive D o7 = false

## See also

• posetJoin -- determines the join for two elements of a poset
• posetMeet -- determines the meet for two elements of a poset
• isLattice -- determines if a poset is a lattice

## Ways to use isDistributive :

• "isDistributive(Poset)"

## For the programmer

The object isDistributive is .