Inflation of Finite Lattices along All-or-nothing Sets Tristan Holmes

We introduce a new generalization of Alan Day’s doubling construction. For ordered sets L and K and a subset E ⊆ ≤L we define the ordered set L ?E K arising from inflation of L along E by K. Under the restriction that L and K are finite lattices, we find those subsets E ⊆ ≤L such that the ordered set L ?E K is a lattice. Finite lattices that can be constructed in this way are classified in terms of their congruence lattices. A finite lattice is binary cut-through codable if and only if there exists a 0 − 1 spanning chain {θi : 0 ≤ i ≤ n} in Con(L) such that the cardinality of the largest block of θi/θi−1 is two for every i with 1 ≤ i ≤ n. These are exactly the lattices that can be obtained from the one element lattice using a sequence of inflations by the two element lattice. We investigate the structure of binary cut-through codable lattices and describe an infinite class of lattices that generate binary cut-through codable varieties.