Mumbai University > Computer Engineering > Sem 3 > Discrete Structures

Marks: 8 Marks

Year: Dec 2014

Distributive lattices are less general than modular lattices but still include some important examples.

Distributive lattices are defined by the following equivalent properties.

Proposition 1 In a lattice L, the distributive conditions

1. x V (y ^ z)=(x V y) ^(x V z) for all x, y, z € L,
2. x ^ (y V z)=(x ^ y) V (x ^ z) for all x, y, z € L,

are equivalent and imply modularity.

A lattice is distributive when it satisfies the equivalent conditions in Proposition 1, For example, the lattice $2^x$ of all subsets of a set X is distributive; so is every sublattice of $2^x$