The class 𝒦 of all
(J,M,m)-extensions of a Boolean algebra 𝒜 can be partially ordered and always
contains a maximum and a minimal element, with respect to this partial ordering.
However, it need not contain a smallest element. Should 𝒦 contain a smallest
element, then 𝒦 has the structure of a complete lattice. Necessary and sufficient
conditions under which 𝒦 does contain a smallest element are derived. A Boolean
algebra 𝒜 is constructed for each cardinal m such that the class of all m-extensions of
𝒜 does not contain a smallest element. One implication of this construction is that if
a Boolean algebra 𝒜 is the Boolean product of a least countably many Boolean
algebras, each of which has more than one m-extension, then the class of all
m-extensions of 𝒜 does not contain a smallest element. The construction also has as
implication that neither the class of all (m,0). products nor the class of all
(m,n)-products of an indexed set {𝒜t}t∈T of Boolean algebras need contain a
smallest element.
