Abstract
If V is a variety of lattices and L a free lattice in V on uncountably many generators, then any cofinal sublattice of L generates all of V. On the other hand, any modular lattice without chains of order-type ω+1 has a cofinal distributive sublattice. More generally, if a modular lattice L has a distributive sublattice which is ‘cofinal modulo intervals with ACC’, this may be enlarged to a cofinal distributive sublattice. Examples are given showing that these existence results are sharp in several ways. Some similar results and questions on existence of cofinal sublattices with DCC are noted.
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