This discussion begins with
the problem of whether or not all (m− n) products of an indexed set {At}t∈T of
Boolean algebras can be obtained as m-extensions of a particular algebra ℱn∗. The
construction of ℱn∗ is similar to the construction of the Boolean product of {At}t∈T;
however the 𝒜t are embedded in ℱn∗ in such a way that their images are
n-independent. If there is a cardinal number n′, satisfying n < n′≦ m, then (m−n′)
products are not obtainable in this manner. For the case n = m an example shows
the answer to be negative. It is explained how the class of m-extensions of
ℱn∗ is situated in the class of all (m − n) products of {At}t∈T. A set of
m-representable Boolean algebras is given for which the minimal (m − n)
product is not m-representable and for which there is no smallest (m − n)
product.
