ORDINAL SUMS OF PROJECTIVE LATTICES AND ISOTONE SECTIONS

A lattice L is projective in a variety V of lattices if whenever f: K L is an epimorphism, there is a homomorphism g: L → K such that f(g(a)) = a for all a ∈ L. Note g is one-to-one and ρ = gf: K → K is a retraction of K; that is, ρ is an endomorphism of K and ρ2 = ρ. The sublattice ρ(K) = g(L) of K is isomorphic to L. The image g(L) is a retract of K. In a slight abuse of terminology, we will also say L is a retract of K. A lattice is projective in V if and only if it a retract of a free lattice FV(X) in V. These remarks are well known. Projective lattices (in the variety of all lattices) were characterized in [5, Theorem 5.7] by a conjunction of four conditions; see also [4]. This extended the results of A. Kostinsky [7], B. Jónsson [6], and R. McKenzie [8] who had characterized finitely generated projective lattices. The map g in the definition of projective lattice is an isomorphism and hence an order-preserving transversal, also known as an isotone section. So the question arises which lattices L have the property that every epimorphis