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Gábor Czédli

Publications and source records attributed to Gábor Czédli.

At least 19 recordsLinked to original sources
arXiv2026

Accumulation points of congruence densities of finite lattices

Let $\mathcal W$ be a nontrivial variety of lattices, and let $L$ be a finite lattice in $\mathcal W$. The congruence density of $L$ with respect to $\mathcal W$ is the number of congruences of $L$ divided by the maximum number of congruences of $|L|$-element lattices belonging to $\mathcal W$. We prove that, with respect to the order and multiplication of the real numbers, the set SCD$(\mathcal W)$ of congruence densities of finite members of $\mathcal W$ as well as its topological closure are countably infinite dually well-ordered monoids. We also prove that the set of accumulation points of SCD$(\mathcal W)$ is either a singleton or it is countably infinite; furthermore, it is a singleton if and only if $\mathcal W$ is a subvariety of the variety of modular lattices. This gives a complicated characterization of modularity: a non-singleton lattice $K$ is modular if and only if SCD$(\mathcal V(K))$, where $\mathcal V(K)$ denotes the variety generated by $K$, has only one accumulation point. The class $\mathcal S$ of semimodular lattices is not a variety, but SCD$(\mathcal S)$ is still meaningful; we prove that SCD$(\mathcal S)$ has exactly one accumulation point.

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arXiv2026

Lattices with congruence densities larger than $3/32$

By a 1997 result of R. Freese, an $n$-element lattice has at most $2^{n-1}$ congruences. This motivates us to define the congruence density cd$(L)$ of a finite $n$-element lattice as $|$Con$(L)|/2^{n-1}$, where $|$Con$(L)|$ is the number of elements of the congruence lattice Con$(L)$ of $L$. We prove that whenever $L$ is a finite lattice with cd$(L)>3/32$, then $L$ has the same number of join-irreducible and meet-irreducible elements. This result is sharp, since there exists a six-element lattice $R_6$ with cd$(R_6)=3/32$ but fewer join-irreducible than meet-irreducible elements. By R. Freese, C. Mureşan, J. Kulin, and the present author's results, lattices with congruence densities larger than $1/8$ have already been described. Here we decrease the lower threshold from $1/8$ to $3/32$. That is, we describe all finite lattices $L$ such that cd$(L)>3/32$. As a corollary, we give the $k$th largest number of congruences of $n$-element lattices for $n>8$ and $k\in\{n+1, n+2, n+3,n+4\}$.

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arXiv2024

Atoms in four-element generating sets of partition lattices

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 that each element of Part($n$) with height one or two (in particular, each atom) belongs to a four-element generating set. Furthermore, our construction leads to a concise and easy proof of a 1996 result of the author stating that the lattice of partitions of a countably infinite set is four-generated as a complete lattice. In a recent paper "Generating Boolean lattices by few elements and exchanging session keys", see https://doi.org/10.30755/NSJOM.16637, the author establishes a connection between cryptography and small generating sets of some lattices, including Part($n$). Hence, it is worth pointing out that by combining a construction given here with a recent paper by the author, "Four-element generating sets with block count width at most two in partition lattices", available at https://tinyurl.com/czg-4gw2, we obtain many four-element generating sets of Part($n$).

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arXiv2024

Four generators of an equivalence lattice with consecutive block counts

The block count of an equivalence $μ\in$ Equ$(A)$ is the number blnum$(μ)$ of blocks of (the partition corresponding to) $μ$. We say that $X=\{μ_1,μ_2,μ_3,μ_4\}$ is a four-element generating set of Equ$(A)$ with consecutive block counts if $X$ generates Equ$(A)$ and blnum$(μ_{i+1})$ = blnum$(μ_{1})+i$ for $i\in\{1,2,3\}$. We prove that if the number of elements of a finite set $A$ is six or at least eight, then Equ$(A)$ has a four-element generating set with consecutive block counts. Also, we present a historical remark on the connection between equivalence lattices and quasiorder lattices.

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arXiv2024

Duality for pairs of upward bipolar plane graphs and submodule lattices

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 set of $G$ and that of $H$, respectively; their role is to define, for each edge of $H$, the corresponding edge of $G$. Let $b$ be an element of an Abelian group $\mathbb A$. An $n$-tuple $(a_1$, $\dots$, $a_n)$ of elements of $\mathbb A$ is a solution of the paired-bipolar-graphs problem $P:=(G,H$, $ϕ,ψ$, $\mathbb A, b)$ if whenever $a_i$ is the ``all-or-nothing-flow'' capacity of the edge $ϕ(i)$ for $i=1, \dots, n$ and $\vec e$ is a maximal directed path of $H$, then by fully exploiting the capacities of the edges corresponding to the edges of $\vec e$ and neglecting the rest of the edges of $G$, we have a flow process transporting $b$ from the source (vertex) of $G$ to the sink of $G$. Let $P':=(H',G'$, $ψ',ϕ'$, $\mathbb A, b)$, where $H'$ and $G'$ are the ``two-outer-facet'' duals of $H$ and $G$, respectively, and $ψ'$ and $ϕ'$ are defined naturally. We prove that $P$ and $P'$ have the same solutions. This result implies George Hutchinson's self-duality theorem on submodule lattices.

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arXiv2024

Large filters of quasiorder lattices can be generated by few elements

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, then $F$ can be generated by few elements.

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arXiv2024

Generating subspace lattices, their direct products, and their direct powers

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 the same holds over every 1- or 2-generated field; in particular, over every field that is a finite degree extension of its prime field. Furthermore, let $F$, $t$, $V$, $d\geq 3$, $[d/2]$, and $m$ denote an arbitrary field, the minimum cardinality of a generating set of $F$, a finite dimensional vector space over $F$, the dimension (assumed to be at least $3$) of $V$, the integer part of $d/2$, and the least cardinal such that $m[d^2/4]$ is at least $t$, respectively. We prove that Sub$(V)$ is $(4+m)$-generated but none of its generating sets is of size less than $m$. Moreover, the $k$-th direct power of Sub$(V)$ is $(5+m)$-generated for many positive integers $k$; for all positive integers $k$ if $F$ is infinite. Finally, let $n$ be a positive integer. For $i=1,\dots, n$, let $p_i$ be a prime number or 0, and let $V_i$ be the 3-dimensional vector space over the prime field of characteristic $p_i$. We prove that the direct product of the lattices Sub$(V_1)$, ..., Sub$(V_n)$ is 4-generated if and only if each of the numbers $p_1$, ..., $p_n$ occurs at most four times in the sequence $p_1$, ..., $p_n$. Neither this direct product nor any of the subspace lattices Sub$(V)$ above is 3-generated.

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arXiv2023

Minimum-sized generating sets of the direct powers of free distributive lattices

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. We determine Gm(FD$(r)^k$) for many pairs $(r,k)$ either exactly or with good accuracy by giving a lower estimate that becomes an upper estimate if we increase it by 1. For example, for $(r,k)=(5,25\,000)$ and $(r,k)=(20,\ 1.489\cdot 10^{1789})$, Gm(FD$(r)^k$) is $300$ and $6000$, respectively. To reach our goal, we give estimates for the maximum number of pairwise unrelated copies of some specific posets (called full segment posets) in the subset lattice of an $n$-element set. In addition to analogous earlier results in lattice theory, a connection with cryptology is also mentioned among the motivations.

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arXiv2023

Generating Boolean lattices by few elements and exchanging session keys

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 $n$-element set) is the least number $k$ such that $n\leq$ Sp($k$). Based on this fact and our 2021 protocol based on equivalence lattices, we outline a cryptographic protocol for exchanging session keys, that is, frequently changing secondary keys. In the present paper, which belongs mainly to lattice theory, we do not elaborate and prove those details of this protocol that modern cryptology would require to guarantee security; the security of the protocol relies on heuristic considerations. However, as a first step, we prove that if an eavesdropper could break every instance of an easier protocol in polynomial time, then P would equal NP. As a byproduct, it turns out that in each nontrivial finite lattice that has a prime filter, in particular, in each nontrivial finite Boolean lattice, the solvability of systems of equations with constant-free left sides but constant right sides is an NP-complete problem.

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arXiv2023

Sperner theorems for unrelated copies of some partially ordered sets in a powerset lattice and minimum generating sets of powers of distributive lattices

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 singleton poset, then Sp$(U,n)$ was determined by E. Sperner in 1928; this result is well known in extremal combinatorics. Later, exactly or asymptotically, Sperner's theorem was extended to other posets by A. P. Dove, J. R. Griggs, G. O. H. Katona, D Nagy, J. Stahl, and W. T. Jr. Trotter. We determine Sp$(U,n)$ for all finite posets with 0 and 1, and we give reasonable estimates for the ``V-shaped'' 3-element poset and the 4-element poset with 0 and three maximal elements. For a lattice $L$, let Gmin($L$) denote the minimum size of generating sets of $L$. We prove that if $U$ is the poset of the join-irreducible elements of a finite distributive lattice $D$, then the function $k\mapsto$ Gmin($D^k)$ is the left adjoint of the function $n\mapsto$ Sp$(U,n)$. This allows us to determine Gmin($D^k)$ in many cases. E.g., for a 5-element distributive lattice $D$, Gmin($D^{2023})=18$ if $D$ is a chain and Gmin($D^{2023})=15$ otherwise. It follows that large direct powers of small distributive lattices are appropriate for our 2021 cryptographic authentication protocol.

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arXiv2023

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

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 irreducible elements and these two elements are complements of each other. A finite poset $P$ is said to be JConSPS-representable if there is an SPS lattice $L$ such that $P$ is isomorphic to the poset J(Con $L$) of join-irreducible congruences of $L$. We prove that if $1<n\in N$ and $P$ is an $n$-element JConSPS-representable poset, then there exists a slim rectangular lattice $L$ such that J(Con $L$) is isomorphic to $P$, the length of $L$ is at most $2n^2$, and $|L|\leq 4n^4$. This offers an algorithm to decide whether a finite poset $P$ is JConSPS-representable (or a finite distributive lattice is ``ConSPS-representable"). This algorithm is slow as G. Czédli, T. Dékány, G. Gyenizse, and J. Kulin proved in 2016 that there are asymptotically $(k-2)!\cdot e^2/2$ many slim rectangular lattices of a given length $k$, where $e$ is the famous constant $\approx 2.71828$. The known properties and constructions of JConSPS-representable posets can accelerate the algorithm; we present a new construction.

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arXiv2023

$\mathcal C_1$-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. While $F/α$ is not even a Hasse diagram in general, we prove that whenever $L$ is an SR lattice and $F$ is a $\mathcal C_1$-diagram of $L$, then $F/α$ is a $\mathcal C_1$-diagram of $L/α$, which is an SR lattice or a chain. The class of lattices isomorphic to the congruence lattices of SPS lattices is closed under taking filters. We prove that this class is closed under two more constructions, which are inverses of taking filters in some sense; one of the two respective proofs relies on an inverse of the quotient diagram construction.

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arXiv2022

Atoms and coatoms in three-generated lattices

In addition to the unique cover $M^+$ of the variety of modular lattices, we also deal with those twenty-three known covers of $M^+$ that can be extracted from the literature. For $M^+$ and for each of these twenty-three known varieties covering it, we determine what the pair formed by the number of atoms and that of coatoms of a three-generated lattice belonging to the variety in question can be. Furthermore, for each variety $W$ of lattices that is obtained by forming the join of some of the twenty-three varieties mentioned above, that is, for $2^{23}$ possible choices of $W$, we determine how many atoms a three-generated lattice belonging to $W$ can have. The greatest number of atoms occurring in this way is only six. In order to point out that this need not be so for larger varieties, we construct a $47\,092$-element three-generated lattice that has exactly eighteen atoms. In addition to purely lattice theoretical proofs, which constitute the majority of the paper, some computer-assisted arguments are also presented.

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arXiv2022

Notes on congruence lattices and lamps of slim semimodular lattices

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 addition to distributivity, there have been seven known properties of the congruence lattices of these lattices. The first two properties were proved by G. Grätzer, the next four by the present author, while the seventh was proved jointly by G. Grätzer and the present author. Five out of the seven properties were found and proved by using lamps, which are lattice theoretic tools introduced by the present author in a 2021 paper. Here, using lamps, we present infinitely many new properties. Lamps also allow us to strengthen the seventh previously known property, and they lead to an algorithm of exponential time to decide whether a finite distributive lattice can be represented as the congruence lattice of an SPS lattice. Some new properties of lamps are also given.

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arXiv2022

Lattices with lots of congruence energy

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 writing, the MathSciNet search for "Title=(graph energy) AND Review Text=(eigenvalue)" returns 351 publications, most of which going after Gutman's definition. A congruence $α$ of a finite algebra $A$ turns $A$ into a simple graph: we connect $x\neq y\in A$ by an edge iff $(x,y)\inα$; we let En$(α)$ be the energy of this graph. We introduce the congruence energy CE$(A)$ of $A$ by CE$(A):=\sum\{$En$(α): α\in$ Con$(A)\}$. Let LAT$(n)$ and CDA$(n)$ stand for the class of $n$-element lattices and that of $n$-element congruence distributive algebras of any type. For a class $\mathcal X$, let CE$(\mathcal X):= \{$CE$(A): A\in \mathcal X\}$. We prove the following. (1) For $α\in A$, En$(α)/2$ is the height of $α$ in the equivalence lattice of $A$. (2) The largest number and the second largest number in CE(LAT($n$)) are $(n-1)\cdot 2^{n-1}$ and, for $n\geq 4$, $(n-1)\cdot 2^{n-2}+2^{n-3}$; these numbers are only witnessed by chains and lattices with exactly one two-element antichain, respectively. (3) The largest number in CE(CDA($n$)) is also $(n-1)\cdot 2^{n-1}$, and if CE$(A)=(n-1)\cdot 2^{n-1}$ for an $A\in$ CDA$(n)$, then Con$(A)$ is a boolean lattice with size $|$Con$(A)|=2^{n-1}$.

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