About Free Lattices and Free Modular Lattices

Ralph Freese Department of Mathematics University of Hawaii at Manoa Honolulu, Hawaii 96822 In this paper we look at some of the problems on free lattices and free modular lattices which are of an order theoretic nature. We review some of the known results, give same new results, and present several open problems. Every countable partially ordered set can be order embedded into a countable free lattice [6]. However, free lattices contain no uncountable chains [25],so the above result does not extend to arbitrary partially ordered sets. The problem of which partially ordered sets can be embedded into a free lattice is open. It is not enough to require that the partially ordered set does not have any uncountable chains. In fact, there are partially ordered sets of height one, which cannot be embedded into any free lattice [23].· The importance of these concepts to projective lattices is discussed. If a > b and there is no a with a > a > b we say a aoVerB b and write ar b. Covers in free lattices and free modular lattices are important to lattice structure theory. We discuss the connec~ion between covers and structure theory and give same of the inore important results about covers. Alan Day has shown that every quotient sublattice (i.e. interval) of a finitely generated free lattice contains a covering [9]. R.A. Dean, on the other hand, has some results on noncovers in free lattices. The analogous problems for free modular lattices are open. Suppose w(x , ... ,x ) is a lattice word and L is a lattice. 1 n _ If we replace all but one of the variables with fixed elements from L we obtain a function f(x) = w(x,a2 , ••• ,an ) fram L to L.