Inflation of Finite Lattices along All-or-Nothing Sets

Abstract

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 lattices and K has maximum and minimum elements, 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 classi ed 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 {f i : 0 i n} in Con(L) such that the cardinality of the largest block of i= i􀀀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. The structure of binary cut-through codable lattices is studied, and those lattices that generate varieties that are binary cut-through codable are characterized.