arXiv subjects
Publications and source records attributed to G. Grätzer.
We prove that every finite distributive lattice $D$ can be represented as the congruence lattice of a rectangular lattice $K$ in which all congruences are principal. We verify this result in a stronger form as an extension theorem.
We characterize the order of principal congruences of a bounded lattice (also of a complete lattice and of a lattice of length 5) as a bounded ordered set. We also state a number of open problems in this new field.
This paper has been withdrawn by the authors due to a crucial computational error. In this paper we deal with the finite case. We prove that a finite bounded ordered set can be represented as the order of principal congruences of a finite \emph{semimodular lattice}.
In a 1998 paper with H. Lakser, the authors proved that every finite distributive lattice $D$ can be represented as the congruence lattice of a finite \emph{semimodular lattice}. Some ten years later, the first author and E. Knapp proved a much stronger result, proving the representation theorem for \emph{rectangular lattices}. In this note we present a short proof of these results.
Let (L_i : i\in I) be a family of lattices in a nontrivial lattice variety V, and let ϕ_i: L_i --> M, for i\in I, be isotone maps (not assumed to be lattice homomorphisms) to a common lattice M (not assumed to lie in V). We show that the maps ϕ_i can be extended to an isotone map ϕ: L --> M, where L is the free product of the L_i in V. This was known for V the variety of all lattices (Yu. I. Sorkin 1952). The above free product L can be viewed as the free lattice in V on the partial lattice P formed by the disjoint union of the L_i. The analog of the above result does not, however, hold for the free lattice L on an arbitrary partial lattice P. We show that the only codomain lattices M for which that more general statement holds are the complete lattices. On the other hand, we prove the analog of our main result for a class of partial lattices P that are not-quite-disjoint unions of lattices. We also obtain some results similar to our main one, but with the relationship lattices:orders replaced either by semilattices:orders or by lattices:semilattices. Some open questions are noted.
We construct all planar semimodular lattices in three simple steps from the direct product of two chains.