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

✍ Scribed by G. Grätzer; E.T. Schmidt

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 328 KB
• Volume: 269
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00501-5

✦ Synopsis

Let L be a bounded lattice, let [a, b] and [c, d] be intervals of L, and let ϕ : [a, b] → [c, d] be an isomorphism between these two intervals. Let us consider the algebra L ← → ϕ = L; ∧, ∨, ϕ, ϕ -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 ← → ϕ 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.