
18 Chapter 1 First concepts
1.2 Complete lattices and complete subnets
We begin our discussion of lattices with the following definition.
Definition 1.7. Let fM , g be an ordered set. Then
(O5) M is called a lattice if for any two elements a, b 2 M ,theinffa, bg and the
supfa, bg exist;
(O6) M is called conditionally complete if for every nonempty, bounded subset S
M ,theinfS and the sup S exist;
(O7) M is called completely ordered or a complete lattice if every subset S M
has an infimum and a supremum.
Every finite subset S Dfa
1
, a
2
, :::, a
n
g of a lattice has an infimum and a supre-
mum. We prove this statement by induction. By definition any subset