arXiv subjects
Publications and source records attributed to Gábor Czédli.
Let $L$ be a finite $n$-element semilattice. We prove that if $L$ has at least $127\cdot 2^{n-8}$ subsemilattices, then $L$ is planar. For $n>8$, this result is sharp since there is a non-planar semilattice with exactly $127\cdot 2^{n-8}-1$ subsemilattices.
For every natural number $n\geq 5$, we prove that the number of subuniverses of an $n$-element lattice is $2^n$, $13\cdot 2^{n-4}$, $23\cdot 2^{n-5}$, or less than $23\cdot 2^{n-5}$. By a subuniverse, we mean a sublattice or the emptyset. Also, we describe the $n$-element lattices with exactly $2^n$, $13\cdot 2^{n-4}$, or $23\cdot 2^{n-5}$ subuniverses.
Let $L$ be an $n$-element finite lattice. We prove that if $L$ has strictly more than $2^{n-5}$ congruences, then $L$ is planar. This result is sharp, since for each natural number $n\geq 8$, there exists a non-planar lattice with exactly $2^{n-5}$ congruences.
K. Adaricheva and M. Bolat have recently proved that if $U_0$ and $U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $j\in \{0,1,2\}$ and $k\in\{0,1\}$ such that $U_{1-k}$ is included in the convex hull of $U_k\cup(\{A_0,A_1, A_2\}\setminus\{A_j\})$. One could say disks instead of circles. Here we prove the existence of such a $j$ and $k$ for the more general case where $U_0$ and $U_1$ are compact sets in the plane such that $U_1$ is obtained from $U_0$ by a positive homothety or by a translation. Also, we give a short survey to show how lattice theoretical antecedents, including a series of papers on planar semimodular lattices by G. Gratzer and E. Knapp, lead to our result.
By a 1941 result of Ph. M. Whitman, the free lattice FL(3) on three generators includes a sublattice $S$ that is isomorphic to the lattice FL($ω$)=FL($\aleph_0$) generated freely by denumerably many elements. The first author has recently "symmetrized" this classical result by constructing a sublattice $S\cong$ FL($ω)$ of FL(3) such that $S$ is SELFDUALLY POSITIONED in FL(3) in the sense that it is invariant under the natural dual automorphism of Fl(3) that keeps each of the three free generators fixed. Now we move to the furthest in terms of symmetry by constructing a selfdually positioned sublattice $S\cong$ FL$(ω)$ of FL(3) such that every element of $S$ is fixed by all automorphisms of FL(3). That is, in our terminology, we embed FL$(ω)$ into FL(3) in a TOTALLY SYMMETRIC way. Our main result determines all pairs $(κ,λ)$ of cardinals greater than 2 such that FL$(κ)$ is embeddable into FL$(λ)$ in a totally symmetric way. Also, we relax the stipulations on $S\cong$FL$κ$ by requiring only that $S$ is closed with respect to the automorphisms of FL$(λ)$, or $S$ is selfdually positioned and closed with respect to the automorphisms; we determine the corresponding pairs $(κ,λ)$ even in these two cases. We reaffirm some of our calculations with a computer program developed by the first author. This program is for the word problem of free lattices, it runs under Windows, and it is freely available.
Let $K_0$ be a compact convex subset of the plane $\mathbb R^2$, and assume that whenever $K_1\subseteq \mathbb R^2$ is congruent to $K_0$, then $K_0$ and $K_1$ are not crossing in a natural sense due to L. Fejes-Tóth. A theorem of L. Fejes-Tóth from 1967 states that the assumption above holds for $K_0$ if and only if $K_0$ is a disk. In a paper appeared in 2017, the present author introduced a new concept of crossing, and proved that L. Fejes-Tóth's theorem remains true if the old concept is replaced by the new one. Our purpose is to describe the hierarchy among several variants of the new concepts and the old concept of crossing. In particular, we prove that each variant of the new concept of crossing is more restrictive then the old one. Therefore, L. Fejes-Tóth's theorem from 1967 becomes an immediate consequence of the 2017 characterization of circles but not conversely. Finally, a mini-survey shows that this purely geometric paper has precursor in combinatorics and, mainly, in lattice theory.
For an integer $n\geq 2$, let NCSL$(n)$ denote the set of sizes of congruence lattices of $n$-element semilattices. We find the four largest numbers belonging to NCSL$(n)$, provided that $n$ is large enough to ensure that $|$NCSL$(n)|\geq 4$. Furthermore, we describe the $n$-element semilattices witnessing these numbers.
By a twenty year old result of Ralph Freese, an $n$-element lattice $L$ has at most $2^{n-1}$ congruences. We prove that if $L$ has less than $2^{n-1}$ congruences, then it has at most $2^{n-2}$ congruences. Also, we describe the $n$-element lattices with exactly $2^{n-2}$ congruences.
Let $λ$ and $κ$ be cardinal numbers such that $κ$ is infinite and either $2\leq λ\leq κ$, or $λ=2^κ$. We prove that there exists a lattice $L$ with exactly $λ$ many congruences, $2^κ$ many ideals, but only $κ$ many filters. Furthermore, if $λ\geq 2$ is an integer of the form $2^m\cdot 3^n$, then we can choose $L$ to be a modular lattice generating one of the minimal modular nondistributive congruence varieties described by Ralph Freese in 1976, and this $L$ is even relatively complemented for $λ=2$. Related to some earlier results of George Grätzer and the first author, we also prove that if $P$ is a bounded ordered set (in other words, a bounded poset) with at least two elements, $G$ is a group, and $κ$ is an infinite cardinal such that $κ\geq |P|$ and $κ\geq |G|$, then there exists a lattice $L$ of cardinality $κ$ such that (i) the principal congruences of $L$ form an ordered set isomorphic to $P$, (ii) the automorphism group of $L$ is isomorphic to $G$, (iii) $L$ has $2^κ$ many ideals, but (iv) $L$ has only $κ$ many filters.
For a positive integer $n$, an $n$-sided polygon lying on a circular arc or, shortly, an $n$-fan is a sequence of $n+1$ points on a circle going counterclockwise such that the "total rotation" $δ$ from the first point to the last one is at most $2π$. We prove that for $n\geq 3$, the $n$-fan cannot be constructed with straightedge and compass in general from its central angle $δ$ and its central distances, which are the distances of the edges from the center of the circle. Also, we prove that for each fixed $δ$ in the interval $(0, 2π]$ and for every $n\geq 5$, there exists a concrete $n$-fan with central angle $δ$ that is not constructible from its central distances and $δ$. The present paper generalizes some earlier results published by the second author and Á. Kunos on the particular cases $δ=2π$ and $δ=π$.
Let $K_0$ be a compact convex subset of the plane $\mathbb R^2$, and assume that $K_1\subseteq \mathbb R^2$ is similar to $K_0$, that is, $K_1$ is the image of $K_0$ with respect to a similarity transformation $\mathbb R^2\to\mathbb R^2$. Kira Adaricheva and Madina Bolat have recently proved that if $K_0$ is a disk and both $K_0$ and $K_1$ are included in a triangle with vertices $A_0$, $A_1$, and $A_2$, then there exist a $j\in \{0,1,2\}$ and a $k\in\{0,1\}$ such that $K_{1-k}$ is included in the convex hull of $K_k\cup(\{A_0,A_1, A_2\}\setminus\{A_j\})$. Here we prove that this property characterizes disks among compact convex subsets of the plane. Actually, we prove even more since we replace "similar" by "isometric" (also called "congruent"). Circles are the boundaries of disks, so our result also gives a characterization of circles.
Let $Q$ be a subset of a finite distributive lattice $D$. An algebra $A$ represents the inclusion $Q\subseteq D$ by principal congruences if the congruence lattice of $A$ is isomorphic to $D$ and the ordered set of principal congruences of $A$ corresponds to $Q$ under this isomorphism. If there is such an algebra for every subset $Q$ containing $0$, $1$, and all join-irreducible elements of $D$, then $D$ is said to be fully (A1)-representable. We prove that every fully (A1)-representable finite distributive lattice is planar and it has at most one join-reducible coatom. Conversely, we prove that every finite planar distributive lattice with at most one join-reducible coatom is fully chain-representable in the sense of a recent paper of G. Grätzer. Combining the results of this paper with another paper by the present author, it follows that every fully (A1)-representable finite distributive lattice is "fully representable" even by principal congruences of finite lattices. Finally, we prove that every chain-representable inclusion $Q\subseteq D$ can be represented by the principal congruences of a finite (and quite small) algebra.
Motivated by a recent paper of G. Grätzer, a finite distributive lattice $D$ is said to be fully principal congruence representable if for every subset $Q$ of $D$ containing $0$, $1$, and the set $J(D)$ of nonzero join-irreducible elements of $D$, there exists a finite lattice $L$ and an isomorphism from the congruence lattice of $L$ onto $D$ such that $Q$ corresponds to the set of principal congruences of $L$ under this isomorphism. Based on earlier results of G. Grätzer, H. Lakser, and the present author, we prove that a finite distributive lattice $D$ is fully principal congruence representable if and only if it is planar and it has at most one join-reducible coatom. Furthermore, even the automorphism group of $L$ can arbitrarily be stipulated in this case. Also, we generalize a recent result of G. Grätzer on principal congruence representable subsets of a distributive lattice whose top element is join-irreducible by proving that the automorphism group of the lattice we construct can be arbitrary.
Kira Adaricheva and Madina Bolat have recently proved that if $U_0$ and $U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $j\in \{0,1,2\}$ and $k\in\{0,1\}$ such that $U_{1-k}$ is included in the convex hull of $U_k\cup(\{A_0,A_1, A_2\}\setminus\{A_j\})$. We give a short new proof for this result, and we point out that a straightforward generalizaton for shperes fails.
After surveying some known properties of compact convex sets in the plane, we give a two rigorous proofs of the general feeling that supporting lines can be slide-turned slowly and continuously. Targeting a wide readership, our treatment is elementary on purpose.
By H. Strietz, 1975, and G. Czédli, 1996, the complete lattice $Equ(A)$ of all equivalences is four-generated, provided the size $|A|$ is an accessible cardinal. Results of I. Chajda and G. Czédli, 1996, G. Takách, 1996, T. Dolgos, 2015, and J.\ Kulin 2016, show that both the lattice $Quo(A)$ of all quasiorders on $A$ and, for $|A|\leq \aleph_0$, the lattice $Tran(A)$ of all transitive relations on $A$ have small generating sets. Based on complicated earlier constructions, we derive some new results in a concise but not self-contained way.
Finite convex geometries are combinatorial structures. It follows from a recent result of M.\ Richter and L.G.\ Rogers that there is an infinite set $T_{rr}$ of planar convex polygons such that $T_{rr}$ with respect to geometric convex hulls is a locally convex geometry and every finite convex geometry can be represented by restricting the structure of $T_{rr}$ to a finite subset in a natural way. An \emph{almost-circle of accuracy} $1-ε$ is a differentiable convex simple closed curve $S$ in the plane having an inscribed circle of radius $r_1>0$ and a circumscribed circle of radius $r_2$ such that the ratio $r_1/r_2$ is at least $1-ε$. % Motivated by Richter and Rogers' result, we construct a set $T_{new}$ such that (1) $T_{new}$ contains all points of the plane as degenerate singleton circles and all of its non-singleton members are differentiable convex simple closed planar curves; (2) $T_{new}$ with respect to the geometric convex hull operator is a locally convex geometry; (3) as opposed to $T_{rr}$, $T_{new}$ is closed with respect to non-degenerate affine transformations; and (4) for every (small) positive $ε\in\real $ and for every finite convex geometry, there are continuum many pairwise affine-disjoint finite subsets $E$ of $T_{new}$ such that each $E$ consists of almost-circles of accuracy $1-ε$ and the convex geometry in question is represented by restricting the convex hull operator to $E$. The affine-disjointness of $E_1$ and $E_2$ means that, in addition to $E_1\cap E_2=\emptyset$, even $ψ(E_1)$ is disjoint from $E_2$ for every non-degenerate affine transformation $ψ$.
The swing lemma, due to G. Grätzer for slim semimodular lattices and extended by G. Czédli and G. Grätzer for all planar semimodular lattices, describes the congruence generated by a prime interval in an efficient way. Here we present a new proof for this lemma, which is shorter than the earlier two. Also, motivated by the swing lemma and mechanical pinball games with flippers, we construct an online game called Swing lattice game. A computer program realizing this game is available from the authors' websites.