This counterexample has been modified subsequently by Ploščica and Tůma to a direct semilattice construction. For a (∨,0)-semilattice, the larger semilattice R(S) is the (∨,0)-semilattice freely generated by new elements t(a,b,c), for a, b, c in S such that c ≤ a ∨ b, subjected to the only relations c=t(a,b,c) ∨ t(b,a,c) and t(a,b,c) ≤ a. Iterating this construction gives the free distributive extension D(S)=\bigcup(R^n(S)\mid n<\omega) of S. Now, for a set Ω, let L(Ω) be the (∨,0)-semilattice defined by generators 1 and ai,x, for i<2 and x in Ω, and relations a0,x ∨ a1,x=1, for any x in Ω. Finally, put G(Ω)=D(L(Ω)).
