DOUBLING CONSTRUCTIONS IN LATTICE THEORY

This paper examines the simultaneous doubling of multiple intervals of a lattice in great detail. In the case of a finite set of W-failure intervals. it is shown that there in a unique smallest lattice mapping homomorphically onto the original lattice, in which the set of W-failures is removed. A nice description of this new lattice is given. This technique is used to show that every lattice that is a bounded homomorphic image of a free lattice has a projective cover. It is also used to give a sufficient condition for a finitely presented lattice to be weakly atomic and shows that the problem of which finitely presented lattices are finite is closely related to the problem of characterizing those finite lattices with a finite W-cover.