Gábor Czédli
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View article: THE LARGEST AND ALL SUBSEQUENT NUMBERS OF CONGRUENCES OF \(n\)-ELEMENT LATTICES
THE LARGEST AND ALL SUBSEQUENT NUMBERS OF CONGRUENCES OF \(n\)-ELEMENT LATTICES Open
For a positive integer \(n\), let SCL\((n)=\{|\)Con \((L)|: L\) is an \(n\)-element lattice\(\}\) stand for the set of Sizes of the Congruence Lattices of \(n\)-element lattices. The \(k\)-th Largest Number of Congruences of \(n\)-element …
View article: A PAIR OF FOUR-ELEMENT HORIZONTAL GENERATING SETS OF A PARTITION LATTICE
A PAIR OF FOUR-ELEMENT HORIZONTAL GENERATING SETS OF A PARTITION LATTICE Open
Let \(\lfloor x \rfloor\) and \(\lceil x\rceil \) denote the lower integer part and the upper integer part of a real number \(x\), respectively. Our main goal is to construct four partitions of a finite set \(A\) with \(n\geq 7\) elements …
View article: Atoms in Four-Element Generating Sets of Partition Lattices
Atoms in Four-Element Generating Sets of Partition Lattices Open
Since Henrik Strietz’s 1975 paper proving that the lattice Part(𝑛) of all partitions of an 𝑛-element finite set is four-generated, more than half a dozen papers have been devoted to four-element generating sets of this lattice. We prove th…
View article: Four-element Generating Sets with Block Count Widths at Most Two in Partition Lattices
Four-element Generating Sets with Block Count Widths at Most Two in Partition Lattices Open
The partitions of a finite set form a so-called partition lattice. Henrik Strietz proved that this lattice has a four-element generating set; his paper has been followed by a dozen others. Two recent papers of the present author indicate t…
View article: Atoms in four-element generating sets of partition lattices
Atoms in four-element generating sets of partition lattices Open
Since Henrik Strietz's 1975 paper proving that the lattice Part($n$) of all partitions of an $n$-element finite set is four-generated, more than half a dozen papers have been devoted to four-element generating sets of this lattice. We prov…
View article: SPERNER THEOREMS FOR UNRELATED COPIES OF POSETS AND GENERATING DISTRIBUTIVE LATTICES
SPERNER THEOREMS FOR UNRELATED COPIES OF POSETS AND GENERATING DISTRIBUTIVE LATTICES Open
For a finite poset (partially ordered set) \(U\) and a natural number \(n\), let \(S(U,n)\) denote the largest number of pairwise unrelated copies of \(U\) in the powerset lattice (AKA subset lattice) of an \(n\)-element set. If \(U\) is t…
View article: Minimum-sized generating sets of the direct powers of free distributive lattices
Minimum-sized generating sets of the direct powers of free distributive lattices Open
For a finite lattice \(L\), let Gm(\(L\)) denote the least \(n\) such that \(L\) can be generated by \(n\) elements. For integers \(r>2\) and \(k>1\), denote by FD\((r)^k\) the \(k\)-th direct power of the free distributive lattice FD(\(r\…
View article: Duality for pairs of upward bipolar plane graphs and submodule lattices
Duality for pairs of upward bipolar plane graphs and submodule lattices Open
Let $G$ and $H$ be acyclic, upward bipolarly oriented plane graphs with the same number $n$ of edges. While $G$ can symbolize a flow network, $H$ has only a controlling role. Let $ϕ$ and $ψ$ be bijections from $\{1, \dots, n\}$ to the edge…
View article: Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices
Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices Open
Following G. Grätzer and E. Knapp (2007), a slim planar semimodular lattice, SPS lattice for short, is a finite planar semimodular lattice having no $M_3$ as a sublattice. An SPS lattice is a slim rectangular lattice if it has exactly two …
View article: Generating subspace lattices, their direct products, and their direct powers
Generating subspace lattices, their direct products, and their direct powers Open
In 2008, László Zádori proved that the lattice Sub$(V)$ of all subspaces of a vector space $V$ of finite dimension at least 3 over a finite field $F$ has a 5-element generating set; in other words, Sub$(V)$ is 5-generated. We prove that th…
View article: Minimum-sized generating sets of the direct powers of free distributive lattices
Minimum-sized generating sets of the direct powers of free distributive lattices Open
For a finite lattice $L$, let Gm($L$) denote the least $n$ such that $L$ can be generated by $n$ elements. For integers $r>2$ and $k>1$, denote by FD$(r)^k$ the $k$-th direct power of the free distributive lattice FD($r$) on $r$ generators…
View article: Sperner theorems for unrelated copies of some partially ordered sets in a powerset lattice and minimum generating sets of powers of distributive lattices
Sperner theorems for unrelated copies of some partially ordered sets in a powerset lattice and minimum generating sets of powers of distributive lattices Open
For a finite poset (partially ordered set) $U$ and a natural number $n$, let Sp$(U,n)$ denote the largest number of pairwise unrelated copies of $U$ in the powerset lattice (AKA subset lattice) of an $n$-element set. If $U$ is the singleto…
View article: Generating Boolean lattices by few elements and exchanging session keys
Generating Boolean lattices by few elements and exchanging session keys Open
Let Sp($k$) denote the number of the $\lfloor k/2\rfloor$-element subsets of a finite $k$-element set. We prove that the least size of a generating subset of the Boolean lattice with $n$ atoms (or, equivalently, the powerset lattice of an …
View article: Large filters of quasiorder lattices can be generated by few elements
Large filters of quasiorder lattices can be generated by few elements Open
For a poset $(P;\leq)$, the quasiorders (AKA preorders) extending the poset order "$\leq$" form a complete lattice $F$, which is a filter in the lattice of all quasiorders of the set $P$. We prove that if the poset order "$\leq$" is small,…
View article: Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices
Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices Open
Following G. Grätzer and E. Knapp (2007), a slim semimodular lattice, SPS lattice for short, is a finite planar semimodular lattice having no $M_3$ as a sublattice. An SPS lattice is a slim rectangular lattice if it has exactly two doubly …
View article: Revisiting Faigle geometries from a perspective of semimodular lattices
Revisiting Faigle geometries from a perspective of semimodular lattices Open
In 1980, U. Faigle introduced a sort of finite geometries on posets that are in bijective correspondence with finite semimodular lattices. His result has almost been forgotten in lattice theory. Here we simplify the axiomatization of these…
View article: Lattice tolerances and congruences
Lattice tolerances and congruences Open
We prove that a tolerance relation of a lattice is a homomorphic image of a congruence relation.
View article: $\mathcal C_1$-diagrams of slim rectangular semimodular lattices permit quotient diagrams
$\mathcal C_1$-diagrams of slim rectangular semimodular lattices permit quotient diagrams Open
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 st…
View article: Congruence structure of planar semimodular lattices: The General Swing Lemma
Congruence structure of planar semimodular lattices: The General Swing Lemma Open
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.
View article: Notes on congruence lattices and lamps of slim semimodular lattices
Notes on congruence lattices and lamps of slim semimodular lattices Open
Since their introduction by G. Grätzer and E. Knapp in 2007, more than four dozen papers have been devoted to finite slim planar semimodular lattices (in short, SPS lattices or slim semimodular lattices) and to some related fields. In addi…
View article: A Property of Lattices of Sublattices Closed Under Taking Relative Complements and Its Connection to 2-Distributivity
A Property of Lattices of Sublattices Closed Under Taking Relative Complements and Its Connection to 2-Distributivity Open
For a lattice L of finite length n , let RCSub( L ) be the collection consisting of the empty set and those sublattices of L that are closed under taking relative complements. That is, a subset X of L belongs to RCSub( L ) if and only if X…
View article: Lattices with lots of congruence energy
Lattices with lots of congruence energy Open
In 1978, motivated by E. Hückel's work in quantum chemistry, I. Gutman introduced the concept of the energy of a finite simple graph $G$ as the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. At the time of wr…
View article: 2-distributivity and lattices of sublattices closed under taking relative complements
2-distributivity and lattices of sublattices closed under taking relative complements Open
For a modular lattice $L$ of finite length, we prove that the distributivity of $L$ is a sufficient condition while its 2-distributivity is a necessary condition that those sublattices of $L$ that are closed under taking relative complemen…
View article: Cyclic congruences of slim semimodular lattices and non-finite axiomatizability of some finite structures
Cyclic congruences of slim semimodular lattices and non-finite axiomatizability of some finite structures Open
We give a new proof of the fact that finite bipartite graphs cannot be axiomatized by finitely many first-order sentences among finite graphs. (This fact is a consequence of a general theorem proved by L. Ham and M. Jackson, and the counte…
View article: Retracts of rectangular distributive lattices and some related observations
Retracts of rectangular distributive lattices and some related observations Open
By a rectangular distributive lattice we mean the direct product of two non-singleton finite chains. We prove that the retracts (ordered by set inclusion and together with the empty set) of a rectangular distributive lattice $G$ form a lat…
View article: A property of meets in slim semimodular lattices and its application to retracts
A property of meets in slim semimodular lattices and its application to retracts Open
Slim semimodular lattices were introduced by G. Grätzer and E. Knapp in 2007, and they have intensively been studied since then. It is often reasonable to give these lattices by their $\mathcal C_1$-diagrams defined by the author in 2017. …
View article: Four-generated direct powers of partition lattices and authentication
Four-generated direct powers of partition lattices and authentication Open
For an integer $n\geq 5$, H. Strietz (1975) and L. Zádori (1986) proved that the lattice Part$(n)$ of all partitions of $\{1,2,\dots,n\}$ is four-generated. Developing L. Zádori's particularly elegant construction further, we prove that ev…