Congruence-preserving Extensions of Finite Lattices to Isoform Lattices
نویسندگان
چکیده
We call a lattice L isoform, if for any congruence relation Θ of L, all congruence classes of Θ are isomorphic sublattices. In an earlier paper, we proved that for every finite distributive lattice D, there exists a finite isoform lattice L such that the congruence lattice of L is isomorphic to D. In this paper, we prove a much stronger result: Every finite lattice has a congruence-preserving extension to a finite isoform lattice.
منابع مشابه
Congruence-preserving Extensions of Finite Lattices to Semimodular Lattices
We prove that every finite lattice has a congruence-preserving extension to a finite semimodular lattice.
متن کاملCongruence-preserving Extensions of Finite Lattices to Sectionally Complemented Lattices
In 1962, the authors proved that every finite distributive lattice can be represented as the congruence lattice of a finite sectionally complemented lattice. In 1992, M. Tischendorf verified that every finite lattice has a congruence-preserving extension to an atomistic lattice. In this paper, we bring these two results together. We prove that every finite lattice has a congruence-preserving ex...
متن کاملThe Strong Independence Theorem for Automorphism Groups and Congruence Lattices of Finite Lattices Theorem. Let L C and L a Be Nite Lattices, L C \ L a = F0g. Then There Exists
The Independence Theorem for the congruence lattice and the auto-morphism group of a nite lattice was proved by V. A. Baranski and A. Urquhart. Both proofs utilize the characterization theorem of congruence lattices of nite lattices (as nite distributive lattices) and the characterization theorem of auto-morphism groups of nite lattices (as nite groups). In this paper, we introduce a new, stron...
متن کاملRepresenting Homomorphisms of Congruence Lattices as Restrictions of Congruences of Isoform Lattices
Let L1 be a finite lattice with an ideal L2. Then the restriction map is a {0, 1}-homomorphism from ConL1 into ConL2. In 1986, the present authors published the converse. If D1 and D2 are finite distributive lattices, and φ : D1 → D2 is a {0, 1}-homomorphism, then there are finite lattices L1 and L2 with an embedding η of L2 as an ideal of L1, and there are isomorphisms ε1 : ConL1 → D1 and ε2 :...
متن کاملProper Congruence - Preserving Extensions of Lattices
We prove that every lattice with more than one element has a proper congruence-preserving extension.
متن کامل