 TOPICS # 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 .

This entry contributed by Matt Insall (author's link)

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Insall, Matt. "Bounded Lattice." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BoundedLattice.html