Given a poset, the two most fundamental derived operations on the elements of the poset are the join and the meet operations. As we have seen earlier, join and meet operations may not exist for all subsets of the poset. This observation motivates the notion of lattices that occur in many different contexts. Let us recall the definition of a lattice from Chapter 1.
If exists, then we call it a sup semilattice. If exists, then we call it an inf semilattice.
In our definition of a lattice, we have required the existence of 's and 's for sets of size two. This is equivalent to the requirement of the existence of 's and 's for sets of finite size by using induction (see Problem 5.6).
We now ...