A finite lattice L is called slim if no three join-irreducible elements of L form an antichain. Slim lattices are planar. Slim semimodular lattices play the main role in [3], where lattice theory is applied to a purely group theoretical problem. After exploring some easy properties of slim lattices and slim semimodular lattices, we give two visual structure theorems for slim semimodular lattices.

Rectangular lattices are special planar semimodular lattices introduced by G. Grätzer and E. Knapp in 2009. By a patch lattice we mean a rectangular lattice whose weak corners are coatoms. As a sort of gluings, 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 “sewn togeth...

Rough sets are efficient for data pre-processing in data mining. Matroids are based on linear algebra and graph theory, and have a variety of applications in many fields. Both rough sets and matroids are closely related to lattices. For a serial and transitive relation on a universe, the collection of all the regular sets of the generalized rough set is a lattice. In this paper, we use the latt...

We study paths between maximal chains, or “flags,” in finite rank semimodular lattices. Two flags are adjacent if they differ on at most one rank. A path is a sequence of flags in which consecutive flags are adjacent. We study the union of all flags on at least one minimum length path connecting two flags in the lattice. This is a subposet of the original lattice. If the lattice is modular, the...

We prove that every finite distributive lattice can be represented as the congruence lattice of a finite (planar) semimodular lattice.

We prove that every finite lattice has a congruence-preserving extension to a finite semimodular lattice.

In a ranked lattice, we consider two maximal chains, or “flags” to be i-adjacent if they are equal except possibly on rank i . Thus, a finite rank lattice is a chamber system. If the lattice is semimodular, as noted in [9], there is a “Jordan-Hölder permutation” between any two flags. This permutation has the properties of an Sn-distance function on the chamber system of flags. Using these noti...

The lattice of closed subsets of a set under such a closure operator is semimodular. Perhaps the best known example of a closure operator satisfying the exchange principle is the closure operator on a vector space W where for X ___ W we let C(X) equal the span of X. The lattice of C-closed subsets of W is isomorphic to Con(W) in a natural way; indeed, if Y _~ W x W and Cg(Y) denotes the congrue...

This paper deals with the relations between graph automorphisms and direct factors of a semimodular lattice of locally finite length.

In this paper we prove that if !.l' is a finite lattice. and r is an integral valued function on !.l' satisfying some very natural then there exists a finite geometric (that is.• semimodular and atomistic) lattice containing asa sublatticesuch that r !.l'restricted to Sf. Moreover. we show that if, for all intervals of. semimodular lattices of length at most r(e) are given. then can be chosen t...