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.