 If we want to disprove Schmidt's problem, the idea is (1) to prove that any generalized Boolean semilattice satisfies URP (which is easy), (2) that URP is preserved under homomorphic image under a weakly distributive homomorphism (which is also easy), and (3) that there exists a distributive (∨,0)-semilattice of cardinality ℵ2 that does not satisfy URP (which is difficult, and uses Kuratowski's Free Set Theorem).Schematically, the construction in the theorem above can be described as follows.For a set Ω, we consider the partially ordered vector space "E(Ω)" defined by generators 1 and "ai,x", for "i<2" and "x" in Ω, and relations "a0,x+a1,x=1", "a0,x ≥ 0", and "a1,x ≥ 0", for any "x" in Ω.