arXiv subjects
Publications and source records attributed to George Grätzer.
For the finite ordered sets $A, D$, write $A^D$ for the ordered set of isotone maps $D \to A$ with the pointwise order. It was proved in earlier work that the order structure of $A^A$ determines~$A$ up to isomorphism. In this note we extend the result to higher function ordered sets such as $A^{(A^A)}$ and $(A^A)^A$. Our main theorem shows that the structure of $A^D$ determines~$A$.
An FN lattice $F$ is a simple, infinite, semidistributive lattice. Its existence was recently proved by R. Freese and J.\,B. Nation. Let $\mathsf{B}_n$ denote the Boolean lattice with $n$ atoms. For a lattice $K$, let $K^+$ denote $K$ with a new unit adjoined. We prove that the finite distributive lattices: $\mathsf{B}_0^+, \mathsf{B}_1^+,\mathsf{B}_2^+, \dots$ can be represented as congruence lattices of infinite semidistributive lattices. The case $n = 0$ is the Freese-Nation result, which is utilized in the proof. We also prove some related representation theorems.
In a 2121 paper with Gábor Czédli, we introduced and verified the Three-pendant Three-crown Property, 3P3C, for congruence lattices of slim, planar, semimodular lattices. The proof is very long; in part, because it relies on Czédli's 2021 paper on lamps. This paper verifies 3P3C using the Swing Lemma, an elementary and short approach.
Let $L$ be a planar semimodular lattice. We call $L$ \emph{slim}, if it has no $\mthree$ sublattice. Let us define an \emph{SPS lattice} as a slim, planar, semimodular lattice $L$. In 2016, I proved a property of congruences of SPS lattices (Two-cover Property) and raised the problem of characterizing them. Since then, more than 50 papers have been published contributing to this problem. In this survey, I provide an overview of this field with major contributions by Gábor Czédli.
This contains Part I of the book: Congruence lattices of finite lattices, which covers about 80 years of research and more than 250 papers.
We prove that a tolerance relation of a lattice is a homomorphic image of a congruence relation.
The Swing Lemma of the second author describes how a congruence spreads from a prime interval to another in a slim (having no $M_3$ sublattice), planar, semimodular lattice. We generalize the Swing Lemma to planar semimodular lattices.
Let $L$ be a slim, planar, semimodular lattice (slim means that it does not contain an ${\mathsf M}_3$-sublattice). We call the interval $I = [o, i]$ of $L$ \emph{rectangular}, if there are complementary $a, b \in I$ such that $a$ is to the left of $b$. We claim that a rectangular interval of a slim rectangular lattice is also a slim rectangular lattice. We will present some applications, including a recent result of G. Czédli. In a paper with E. Knapp about a dozen years ago, we introduced natural diagrams} for slim rectangular lattices. Five years later, G. Czédli introduced ${\E C}_1$-diagrams} We prove that they are the same.
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{$\mathcal{C}_1$-diagrams}. We revisit the concept of \emph{natural diagrams} I introduced with E.~Knapp about a dozen years ago. Given a slim rectangular lattice $L$, we construct its natural diagram in one simple step. \emph{The second result} shows that for a slim rectangular lattice, a~natural diagram is the same as a $\mathcal{C}_1$-diagram. Therefore, natural diagrams have all the nice properties of $\mathcal{C}_1$-diagrams.
Let $L$ be a finite lattice and let $I$ be an ideal of $L$. Then the restriction map is a bounded lattice homomorphism of the congruence lattice of~$L$ into the congruence lattice of $I$. In a 2009 paper, the authors proved the converse. In a 2012 paper, G. Czédli proved an analogous result for rectangular lattices. In this paper, we prove a stronger form of Czédli's result and provide a short, elementary, and direct proof.
A planar semimodular lattice $L$ is \emph{slim} if $\mathbf{M}_3$ is not a sublattice of $L$. In a recent paper, G. Czédli introduced a very powerful diagram type for slim, planar, semimodular lattices. This short note proves the existence of such diagrams.
A planar semimodular lattice $K$ is \emph{slim} if $\mathsf{M}_{3}$ is not a sublattice of~$K$. In a recent paper, G. Czédli found four new properties of congruence lattices of slim, planar, semimodular lattices, including the \emph{No Child Property}: \emph{Let~$\mathcal{P}$ be the ordered set of join-irreducible congruences of $K$. Let $x,y,z \in \mathcal{P}$ and let $z$ be a~maximal element of $\mathcal{P}$. If $x \neq y$ and $x, y \prec z$ in $\mathcal{P}$, then there is no element $u$ of $\mathcal{P}$ such that $u \prec x, y$ in $\mathcal{P}$.} We are applying my Swing Lemma, 2015, and a type of standardized diagrams of Czédli's, to verify his four properties.
For a finite distributive lattice $D$, let us call $Q \subseteq D$ \emph{principal congruence representable}, if there is a finite lattice $L$ such that the congruence lattice of $L$ is isomorphic to $D$ and the principal congruences of $L$ correspond to $Q$ under this isomorphism. We find a necessary condition for representability by principal congruences and prove that for finite distributive lattices with a join-irreducible unit element this condition is also sufficient.
Zilber's Theorem states that a finite lattice $L$ is planar if{}f it has a complementary order relation. We provide a new proof for this crucial result and discuss some applications, including a canonical form for finite planar lattices and an analysis of coverings in the left-right order.
A planar semimodular lattice $K$ is \emph{slim} if $\mathsf{M}_3$ is not a sublattice of~$K$. In a recent paper, G. Czédli found four new properties of congruence lattices of slim, planar, semimodular lattices, including the \emph{No Child Property}: \emph{Let~$P$ be the ordered set of join-irreducible congruences of $K$. Let $x,y,z \in P$ and let $z$ be a~maximal element of $P$. If $x \neq y$, $x, y \prec z$ in $P$, then there is no element $u$ of $P$ such that $u \prec x, y$ in $P$.} We are applying my Swing Lemma, 2015, and a type of standardized diagrams of Czédli's, to verify Czédli's four properties.
The systematic study of planar semimodular lattices started in 2007 with a series of papers by G. Grätzer and E. Knapp. These lattices have connections with group theory and geometry. A planar semimodular lattice $L$ is {\it slim} if $M_3$ it is not a sublattice of $L$. In his 2016 monograph, "The Congruences of a Finite Lattice, A \emph{Proof-by-Picture Approach}", the second author asked for a characterization of congruence lattices of slim, planar, semimodular lattices. In addition to distributivity, both authors have previously found specific properties of these congruence lattices. In this paper, we present a new property, the {\it Three-pendant Three-crown Property}. The proof is based on the first author's papers: 2014 (multifork extensions), 2017 ($\mathcal C_1$-diagrams), and a recent paper (lamps), introducing the tools we need.
Two years ago, I characterized the order $\Princl L$ of principal congruences of a bounded lattice $L$ as a bounded order. If $K$ and $L$ are bounded lattices and $\gf$ is a \zo homomorphism of $K$ into~$L$, then there is a natural isotone \zo-map $\gf_{\Hom}$ from $\Princl K$ into $\Princl L$. We prove the converse: For bounded orders $P$ and $Q$ and an isotone \zo map $\gy$ of $P$ into $Q$, we represent $P$ and $Q$ as $\Princl K$ and $\Princl L$ for bounded lattices $K$ and $L$ with a \zo homomorphism $\gf$ of $K$ into $L$, so that $\gy$ is represented as $\gf_{\Hom}$.
In an earlier paper, to describe how a congruence spreads from a prime interval to another in a finite lattice, I introduced the concept of prime-perspectivity and its transitive extension, prime-projectivity and proved the Prime-projectivity Lemma. In this paper, I specialize the Prime-projectivity Lemma to slim, planar, semimodular lattices to obtain the Swing Lemma, a very powerful description of the congruence generated by a prime interval in this special class of lattices.