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 :[a,b]→[c,d] be an isomorphism between these two intervals. Let us consider the algebra , 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 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.