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...

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.

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

 In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $|J(L)setminus A(L)| leq 3$, where $J(L)$ and $A(L)$ are the set of join-irreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices ha...

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...

A planar (upper) semimodular lattice L is slim if the five-element nondistributive modular M3 does not occur among its sublattices. (Planar lattices are finite by definition.) Slim rectangular as particular were defined G. Grätzer and E. Knapp in 2007. In 2009, they also proved that congruence of with at least three elements same those lattices. order to provide an effective tool for studying t...

Let L be a slim, planar, semimodular lattice (slim means that it does not contain an $${{\textsf{M}}}_3$$ -sublattice). We call the interval $$I = [o, i]$$ of rectangular, if there are complementary $$a, b \in I$$ such is to left b. claim rectangular slim also lattice. will present some applications, including recent result G. Czédli. In paper with E. Knapp about dozen years ago, we introduced ...