8. Distributive Lattices
2011
Every dog must have his day. In this chapter and the next we will look at the two most important lattice varieties: distributive and modular lattices. Let us set the context for our study of distributive lattices by considering varieties generated by a single finite lattice. A variety V is said to be locally finite if every finitely generated lattice in V is finite. Equivalently, V is locally finite if the relatively free lattice FV(n) is finite for every integer n> 0. Theorem 8.1. If L is a finite lattice and V = HSP(L), then Hence HSP(L) is locally finite. |FV(n) | ≤ |L | |L|n Proof. If K is any collection of lattices and V = HSP(K), then FV(X) ∼ = FL(X)/θ where θ is the intersection of all homomorphism kernels ker f such that f: FL(X) → L for some L ∈ K. (This is the technical way of saying that FL(X)/θ satisfies exactly the equations that hold in every member of K.) When K consists of a single finite lattice {L} and |X | = n, then there are |L | n distinct mappings of X into L, and hence |L | n distinct homomorphisms fi: FL(X) → L (1 ≤ i ≤ |L | n). 1 The range of each fi is a sublattice of L. Hence FV(X) ∼ = FL(X)/θ with θ = ⋂ ker fi means that FV(X) is a subdirect product of |L | n sublattices of L, and so a sublattice of the direct product ∏ 1≤i≤|L | n L = L |L|n, making its cardinality at most |L||L|n. 2 □ We should note that not every locally finite lattice variety is generated by a finite lattice. Now it is clear that there is a unique minimum nontrivial lattice variety, viz., the one generated by the two element lattice 2, which is isomorphic to a sublattice of any nontrivial lattice. We want to show that HSP(2) is the variety of all distributive lattices. Lemma 8.2. The following lattice equations are equivalent. (1) x ∧ (y ∨ z) ≈ (x ∧ y) ∨ (x ∧ z) (2) x ∨ (y ∧ z) ≈ (x ∨ y) ∧ (x ∨ z)