Lattices with finite \(W\)-covers (Q1272213)

A lattice \(\widetilde L\) is called a finite \(W\)-cover of a finite lattice \(L\) if \(\widetilde L\) satisfies Whitman's condition \(W\) and there is a homomorphism \(f\) of \(\widetilde L\) onto \(L\) such that whenever \(g\) is a homomorphism of a lattice \(R\) satisfying \(W\) onto \(L\) then \(g=f\cdot h,\) where \(h\) is a homomorphism of \(R\) onto \(\widetilde L\). The purpose of the paper is to show that the existence of a finite \(W\)-cover for a given lattice is decidable. It is shown that for a given finite lattice \(L\) and a subset \(A\) of \(L,\) the existence of a finite lattice \(Q\) and homomorphism \(f\) of \(Q\) onto \(L\) such that whenever \(I=[a\wedge b,c\vee d]\) is an interval in \(Q\) then either \(I\cap \{a,b,c,d\}\not =\emptyset \) or \(f(I)\cap A\not=\emptyset \) is decidable. It is proved that for a finite \(W\)-cover \(\widetilde L\) of \(L\), \(|\widetilde L|\leq 86\cdot| L|\).