Discrete Mathematical Structures by Pearson

8-18 Discrete Mathematical Structures

8.14 BOUNDED LATTICES

A lattice L is said to be bounded if it has a least element 0 and a greatest element 1.

The non-negative integers with the usual ordering

0 < 1 < 2 < 3…

has 0 as a lower bound but has no upper bound. The lattice P(U) of all subsets of a universal set

U is a bounded lattice with U as an upper bound and the empty set f as lower bound.

If L ={a

1

, a

2

, ………, a

n

}is a lattice, then a

1

∨ a

2

∨ … ∨ a

n

and a

1

∧ a

2

∧ … ∧ a

n

are upper and

lower bounds of L.

Example 1 Let L = {1, 2, 3, 6} and ≤ be the divisibility relation on L. Hence, (L, ≤ ) is a bounded

lattice as it has least element 1 and greatest ...