40 citations · 89 across the 38 of their papers we have counts for
40 papers
The problem for finite ordered sets
Jonathan David Farley, George Grätzer
For ordered sets and , let denote the ordered set of order-preserving maps from to , ordered pointwise. In 1984, D. Duffus asked whether forces $…
Generalizing the Problem for Finite Ordered Sets
George Grätzer
Let X and Y be finite ordered sets and let denote the ordered set of order-preserving maps from to . Let be a~term formed from a~single variable by exponentiation.…
Multi-algebras as tolerance quotients of algebras
G. Grätzer, R. Quackenbush
If is an algebra and \bgt is a tolerance on , then $A/\bgt$ is a multi-algebra in a natural way. We give an example to show that not every multi-algebra arises in this manne…
On the algorithmic construction of the 1960 sectional complement
G. Grätzer, G. Klus, A. Nguyen
In 1960, G. Grätzer and E.\,T. Schmidt proved that every finite distributive lattice can be represented as the congruence lattice of a sectionally complemented finite lattice .…
Lattice tolerances and congruences
Gábor Czédli, George Grätzer
We prove that a tolerance relation of a lattice is a homomorphic image of a congruence relation.
Notes on planar semimodular lattices. VI. On the structure theorem of planar semimodular lattices
G. Grätzer
In a recent paper, G. Czédli and E.\,T. Schmidt present a structure theorem for planar semimodular lattices. In this note, we present an alternative proof.
Homomorphisms and principal congruences of bounded lattices. III. The Independence Theorem
G. Grätzer
A new result of G. Czédli states that for an ordered set with at least two elements and a group , there exists a bounded lattice such that the ordered set of principal c…
Congruence structure of planar semimodular lattices: The General Swing Lemma
Gábor Czédli, George Grätzer, Harry Lakser
The Swing Lemma of the second author describes how a congruence spreads from a prime interval to another in a slim (having no sublattice), planar, semimodular lattice. We gen…
Homomorphisms and principal congruences of bounded lattices. II. Sketching the proof for sublattices
G. Grätzer
A recent result of G. Czédli relates the ordered set of principal congruences of a bounded lattice with the ordered set of principal congruences of a~bounded sublattice of…
On slim rectangular lattices
George Grätzer
Let be a slim, planar, semimodular lattice (slim means that it does not contain an -sublattice). We call the interval of \emph{rectangular}, if…
On a property of congruence lattices of slim, planar, semimodular lattices
George Grätzer
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…
Another research note
George Grätzer
Let be a slim, planar, semimodular lattice (slim means that it does not contain -sublattices). We call the interval of \emph{rectangular}, if th…
A rectangular interval of a rectangular lattice is a rectangular lattice
G. Grätzer
Let be a slim, planar, semimodular lattice (slim means that it does not contain -sublattices). We call the interval of \emph{rectangular}, if th…
The Congruences of a Finite Lattice, A "Proof-by-Picture" Approach, third edition
George Grätzer
This contains Part I of the book: Congruence lattices of finite lattices, which covers about 80 years of research and more than 250 papers.
Homomorphisms and principal congruences of bounded lattices
George Grätzer
Two years ago, I characterized the order $\Princl L$ of principal congruences of a bounded lattice as a bounded order. If and are bounded lattices and $\gf$ is a \zo ho…
Congruences in slim, planar, semimodular lattices: The Swing Lemma
George Grätzer
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 e…
Congruences and trajectories in planar semimodular lattices
George Grätzer
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~$\fq$ (\emp…
On a result of Gábor Czédli concerning congruence lattices of planar semimodular lattices
George Grätzer
A planar semimodular lattice is slim if it does not contain as a sublattice. An SPS lattice is a slim, planar, semimodular lattice. A recent result of Gábor Czédli proves tha…
Congruences and prime-perspectivities in finite lattices
G. Grätzer
IIn a finite lattice, a congruence spreads from a prime interval to another by a sequence of congruence-perspectivities through \emph{intervals of arbitrary size}, by a 1955 result…
Isoform lattices
G. Grätzer, E. T. Schmidt, R. W. Quackenbush
Let be a lattice. We call a congruence relation $\gQ$ of isoform, if any two congruence classes of $\gQ$ are isomorphic (as lattices). Let us call the lattice isoform,…
Representing finite distributive lattices as congruence lattices of lattices
George Grätzer
Dilworth's theorem. Every finite distributive lattice can be represented as the congruence lattice of a finite lattice . We want: Every finite distributive lattice can b…
The order of principal congruences of a bounded lattice
G. Grätzer
We characterize the order of principal congruences of a bounded lattice as a bounded ordered set. We also state a number of open problems in this new field.
A technical lemma for congruences of finite lattices
George Grätzer
The classical Technical Lemma for congruences is not difficult to prove but it is very efficient in its applications. We present here a Technical Lemma for congruences on \emph{fin…
Congruences of fork extensions of slim semimodular lattices
George Grätzer
For a slim, planar, semimodular lattice and covering square~, G.~Czédli and E.\,T.~Schmidt introduced the fork extension, , which is also a slim, planar, semimodular l…