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
.