Projective Ordinal Sums of Lattices and Isotone Sections

This note gives a complete characterization of when the ordinal sum of two lattices (the lattice obtained by placing the second lattice on top of the first) is projective. This characterization applies not only to the class of all lattices, but to any variety of lattices, and in particular, to the class of distributive lattices. Lattices L with the property that every epimorphism onto L has an isotone section are also characterized. In this paper we will determine when an ordinal sum of two lattices is projective in an arbitrary variety. The definition of projectivity most useful for the paper is the following one: A lattice L is projective in a variety V of lattices if L ∈ V and whenever K ∈ V and 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 [?, Theorem 5.7] by a conjunction of four conditions; see also [?]. This extended the results of A. Kostinsky [?], B. Jónsson [?], and R. McKenzie [?] 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 to ker f , also known as an isotone section. So the question arises which lattices L have the property that every epimorphism K L has an isotone section? We will show that a lattice L has this property if and only if it satisfies the following condition. For each a ∈ L there are two finite sets A(a) ⊆ {x ∈ L : x ≥ a} and B(a) ⊆ {x ∈ L : x ≤ a} such that if a ≤ b then A(a) ∩ B(b) 6= ∅. (A for ‘above’; B for ‘below.’) A lattice, or more generally a partially ordered set, satisfying this condition is called finitely separable. Date: June 6, 2014. 1991 Mathematics Subject Classification. 06B25, 06B05.