Representing congruence lattices of lattices with partial unary operations as congruence lattices of lattices.: I.: Interval equivalence

Let L be a bounded lattice, let [a, b] and [c, d] be intervals of L, and let phi:[a, b] --> [c, d] be an isomorphism between these two intervals. Let us consider the algebra L (phi) over left right arrow = (L; boolean AND,boolean OR, phi,phi(-1)), which is a lattice with two partial unary operations. We construct a bounded lattice K (in fact, a convex extension of L) such that the congruence lattice of L (phi) over left right arrow is isomorphic to the congruence lattice of K, and extend this result to (many) families of isomorphisms. This result presents a lattice K whose congruence lattice is derived from the congruence lattice of L in a novel way. (C) 2003 Elsevier Inc. All rights reserved.