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Bounded Lattice

A bounded lattice is an algebraic structure L=(L, ^ , v ,0,1), such that (L, ^ , v ) is a lattice, and the constants 0,1 in L satisfy the following:

1. for all x in L, x ^ 1=x and x v 1=1,

2. for all x in L, x ^ 0=0 and x v 0=x.

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

There is a natural relationship between bounded lattices and bounded lattice-ordered sets. In particular, given a bounded lattice, (L, ^ , v ,0,1), the lattice-ordered set (L,<=) that can be defined from the lattice (L, ^ , v ) is a bounded lattice-ordered set with upper bound 1 and lower bound 0. Also, one may produce from a bounded lattice-ordered set (L,<=) a bounded lattice (L, ^ , v ,0,1) 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 (L,<=) can have bounded subposets that are also lattice-ordered, but whose bounds are not the same as the bounds of (L,<=); however, any subalgebra of a bounded lattice L=(L, ^ , v ,0,1) is a bounded lattice with the same upper bound and the same lower bound as the bounded lattice L.

For example, let X={a,b,c}, and let L=(L, ^ , v ,0,1) be the power set of X, considered as a bounded lattice:

1. L={emptyset,{a},{b},{c},{a,b},{a,c},{b,c},X}

2. 0=emptyset and 1=X

3. ^ is union: for A,B in L, A v B=A union B

4. v is intersection: for A,B in L, A ^ B=A intersection B.

Let Y={a,b}, and let K=(K, ^ , v ,0^',1^') be the power set of Y, also considered as a bounded lattice:

1. K={emptyset,{a},{b},Y}

2. 0^'=emptyset and 1^'=Y

3. ^ is union: for A,B in L, A ^ B=A union B

4. v is intersection: for A,B in L, A v B=A intersection B.

Then the lattice-ordered set (K,<=) that is defined by setting A<=B iff A subset= B is a substructure of the lattice-ordered set (L,<=) that is defined similarly on L. Also, the lattice (K, ^ , v ) is a sublattice of the lattice (l, ^ , v ). However, the bounded lattice K=(K, ^ , v ,0^',1^') is not a subalgebra of the bounded lattice L=(L, ^ , v ,0,1), precisely because 1!=1^'.

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

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Cite this as:

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

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