Bounded Lattice

A bounded lattice is an algebraic structure , such that  is a lattice, and the constants  satisfy the following:

1. for all ,  and ,

2. for all ,  and .

The element 1 is called the upper bound, or top of  and the element 0 is called the lower bound or bottom of .

There is a natural relationship between bounded lattices and bounded lattice-ordered sets. In particular, given a bounded lattice, , the lattice-ordered set  that can be defined from the lattice  is a bounded lattice-ordered set with upper bound 1 and lower bound 0. Also, one may produce from a bounded lattice-ordered set  a bounded lattice  in a pedestrian manner, in essentially the same way one obtains a lattice from a lattice-ordered set. Some authors do not distinguish these structures, but here is one fundamental difference between them: A bounded lattice-ordered set  can have bounded subposets that are also lattice-ordered, but whose bounds are not the same as the bounds of ; however, any subalgebra of a bounded lattice  is a bounded lattice with the same upper bound and the same lower bound as the bounded lattice .

For example, let , and let  be the power set of , considered as a bounded lattice:

1. 

2.  and 

3.  is union: for , 

4.  is intersection: for , .

Let , and let  be the power set of , also considered as a bounded lattice:

1. 

2.  and 

3.  is union: for , 

4.  is intersection: for , .

Then the lattice-ordered set  that is defined by setting  iff  is a substructure of the lattice-ordered set  that is defined similarly on . Also, the lattice  is a sublattice of the lattice . However, the bounded lattice  is not a subalgebra of the bounded lattice , precisely because .