E. Tamás Schmidt

Slim Semimodular Lattices. I. A Visual Approach

A finite lattice L is called slim if no three join-irreducible elements of L form an antichain. Slim lattices are planar. After exploring some elementary properties of slim lattices and slim semimodular lattices, we give two visual structure theorems for slim semimodular lattices.

Slim Semimodular Lattices. II. A Description by Patchwork Systems

Rectangular lattices are special planar semimodular lattices introduced by G. Grätzer and E. Knapp in Acta Sci Math 75:29–48, 2009. A patch lattice is a rectangular lattice whose weak corners are coatoms. As a variant of gluing, we introduce the concept of a patchwork system. We prove that every glued sum indecomposable, planar, semimodular lattice is a patchwork of its maximal patch lattice intervals. For a planar modular lattice, our patchwork system is the same as the S-glued system introduced by C. Herrmann in Math Z 130:255–274, 1973. Among planar semimodular lattices, patch lattices are characterized as the patchwork-irreducible ones. They are also characterized as the indecomposable ones with respect to gluing over chains; this gives another structure theorem.

On the Scope of Averaging for Frankl's Conjecture

Let

Cover-preserving embeddings of finite length semimodular lattices into simple semimodular lattices

\mathcal F

The Jordan-Hölder theorem with uniqueness for groups and semimodular lattices

be a union-closed family of subsets of an m-element set A. Let

Composition series in groups and the structure of slim semimodular lattices

n=|{\mathcal F}|\ge 2

CD-independent subsets in distributive lattices

. For b ∈ A let w(b) denote the number of sets in

A cover-preserving embedding of semimodular lattices into geometric lattices

\mathcal F

Finite distributive lattices are congruence lattices of almost-geometric lattices

containing b minus the number of sets in

Congruence lattices of p -algebras

\mathcal F