A rectangular interval of a rectangular lattice is a rectangular lattice

Diagrams and rectangular extensions of planar semimodular lattices

In 2009, G. Grätzer and E. Knapp proved that every planar semimodular lattice has a rectangular extension. We prove that, under reasonable additional conditions, this extension is unique. This theorem naturally leads to a hierarchy of special diagrams of planar semimodular lattices. Besides that these diagrams are unique in a strong sense, we explore many of their further properties. Finally, we demonstrate the power of our new diagrams in two ways. First, we prove a simplified version of our earlier Trajectory Coloring Theorem, which describes the inclusion con(p)\supseteq\con(q) for prime in…

Applying the Czédli-Schmidt Sequences to congruence properties of planar semimodular lattices

Following G.~Grätzer and E.~Knapp, 2009, a planar semimodular lattice $L$ is \emph{rectangular}, if~the left boundary chain has exactly one doubly-irreducible element, $c_l$, and the right boundary chain has exactly one doubly-irreducible element, $c_r$, and these elements are complementary. The Czédli-Schmidt Sequences, introduced in 2012, construct rectangular lattices. We use them to prove some structure theorems. In particular, we prove that for a slim (no $\mathsf{M}_3$ sublattice) rectangular lattice~$L$, the congruence lattice $\Con L$ has exactly $\length[c_l,1] + \length[c_r,1]$ dual …

Another research note

Let $L$ be a slim, planar, semimodular lattice (slim means that it does not contain ${\mathsf M}_3$-sublattices). We call the interval $I = [o, i]$ of $L$ \emph{rectangular}, if there are $u_l, u_r \in [o, i] - \{o,i\}$ such that $i = u_l \vee u_r$ and $o = u_l \wedge u_r$ where $u_l$ is to the left of $u_r$. \emph{The first result}: a rectangular interval of a rectangular lattice is a rectangular lattice. As an application, we get a recent result of G. Czédli. In a 2017 paper, G. Czédli introduced a very powerful diagram type for slim, planar, semimodular lattices, the \emph{$\mathcal{C}_1$-d…

$\mathcal C_1$-diagrams of slim rectangular semimodular lattices permit quotient diagrams

Slim semimodular lattices (for short, SPS lattices) and slim rectangular lattices (for short, SR lattices) were introduced by G. Grätzer and E. Knapp in 2007 and 2009. These lattices are necessarily finite and planar, and they have been studied in more then four dozen papers since 2007. They are best understood with the help of their $\mathcal C_1$-diagrams, introduced by the author in 2017. For a diagram $F$ of a finite lattice $L$ and a congruence $α$ of $L$, we define the ``quotient diagram'' $F/α$ by taking the maximal elements of the $α$-blocks and preserving their geometric positions. Wh…

On the coset category of a skew lattice

Skew lattices are non-commutative generalizations of lattices. The coset structure decomposition is an original approach to the study of these algebras describing the relation between its rectangular classes. In this paper we will look at the category determined by these rectangular algebras and the morphisms between them, showing that not all skew lattices can determine such a category. Furthermore, we will present a class of examples of skew lattices in rings that are not strictly categorical, and present sufficient conditions for skew lattices of matrices in rings to constitute $\wedge$-dis…

An extension theorem for planar semimodular lattices

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.

Large intervals in the clone lattice

We give three examples of large intervals in the lattice of (local) clones on an infinite set X, by exhibiting clones C_1, C_2, C_3 such that: (1) the interval [C_1, O] in the lattice of local clones is (as a lattice) isomorphic to {0,1,2, ...} under the divisibility relation, (2) the interval [C_2, O] in the lattice of local clones is isomorphic to the congruence lattice of an arbitrary semilattice, (3) the interval [C_3, O] in the lattice of all clones is isomorphic to the lattice of all filters on X. These examples explain the difficulty of obtaining a satisfactory analysis of the clone lat…

A short proof of the congruence representation theorem for semimodular lattices

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.

在不改变同余格的情况下缩短细长平面半模格的长度

Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices

本文证明任意含 $n$ 个元素的 JConSPS-可表示偏序集都可由长度不超过 $2n^2$、大小不超过 $4n^4$ 的细长矩形格表示，并据此提出可判定算法及一种新构造。

Congruences and trajectories in planar semimodular lattices

A 1955 result of J.~Jakubí k states that for the prime intervals $\fp$ and $\fq$ of a finite lattice, $\con{\fp} \geq \con{\fq}$ if{}f $\fp$ is congruence-projective to~$\fq$ (\emph{via} intervals of arbitrary size). The problem is how to determine whether $\con{\fp} \geq \con{\fq}$ involving only prime intervals. Two recent papers approached this problem in different ways. G. Czédli's used trajectories for slim rectangular lattices---a special subclass of slim, planar, semimodular lattices. I used the concept of prime-projectivity for arbitrary finite lattices. In this note I show how my appr…