Unbounded Semidistributive Lattices

The purpose of this note is to illustrate a construction technique which has proved useful, and apply it to solve an interesting problem. For a more complete discussion of the theory of bounded homomorphisms and lattices, see Chapter II of [3]. 1. Unbounded lattices We begin with a well known criterion for meet semidistributivity. If L is a finite lattice and a ∈ J(L), let κ(a) be the largest element above a * but not above a, if such an element exists. We regard κ : J(L) → M(L) as a partial map. Lemma 1. Let L be a finite lattice. Then L satisfies SD ∧ if and only if κ(a) exists for each a ∈ J(L). Moreover, if a finite lattice L satisfies SD ∧ , then κ maps J(L) onto M(L). If L also satisfies SD ∨ , then κ is one-to-one, and the dual map κ d : M(L) → J(L) is its inverse. We define the standard dependency relations on J(L) as follows (assuming SD ∧ for the first three). a A b if b < a < κ(b) * , a B b if a = b, b κ(a), b * ≤ κ(a), a C b if a A b or a B b, a D b if there exists x ∈ L such that a ≤ b ∨ x but a b * ∨ x. Thus A ∪ B = C ⊆ D. The dual relations are defined on M(L). In semidistributive lattices, they behave particularly nicely, as shown by the following result from [4]. Lemma 2. Let L be a finite semidistributive lattice, and let a, b ∈ J(L). 1. a A b if and only if κ(a) B d κ(b). 2. a B b if and only if κ(a) A d κ(b). Recall the basic results on boundedness (in the sense of McKenzie) and D-cycles. Theorem 3. A finite lattice L is lower bounded if and only if L contains no D-cycle. Moreover, every lower bounded lattice satisfies SD ∨ .