Representing congruence lattices of lattices with partial unary operations as congruence lattices of lattices. II. Interval ordering

In Part I of this paper, we introduced a method of making two isomorphic intervals of a bounded lattice congruence equivalent. In this paper, we make one interval dominate another one. Let L be a bounded lattice, let [a,b] and [c,d] be intervals of L, and let φ be a homomorphism of [a,b] onto [c,d]. We construct a bounded (convex) extension K of L such that a congruence Θ of L has an extension to K iff x≡y(Θ) implies that xφ≡yφ(Θ), for a x y b, in which case, Θ has a unique extension to K. This result presents a lattice K whose congruence lattice is derived from the congruence lattice of L in a new way, different from the one presented in Part I. The main technical innovation is the 2/3-Boolean triple construction, which owes its origin to the Boolean triple construction of G. Grätzer and F. Wehrung.