Let S be a bounded join-semilatticesemilattice without maximal elements. Can we always construct an atomless boolean algebra B, containing S as a subsemilattice, such that S is cofinal in B-{1}? That is, for every x≠1 from B there is y∈S such that x≤y.
More detailed explanation since apparently my terminology was rather ambiguous:
S is a partially ordered set with the least element 0, and every pair of elements from S has an infimum (greatest lower bound). S has no maximal elements, that is for every x∈S there is y∈S greater than x.
If B is an atomless boolean algebra, B-{1} would definitely satisfy the above conditions.
The question is, if it is possible for any semilattice S satisfying the above conditions to cofinally embed it into an atomless boolean algebra B. That is, for any x∈B there is y∈S such that y is greater than x in B.