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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 is join-closed, meet-closed, and whenever {a, x, b} ⊆ S, y ∈ L, x ∧ y = a, and x ∨ y = b, then y ∈ S. We prove that (1) the poset RCSub(L) with respect to set inclusion is lattice of length n + 1, (2) if RCSub(L) is a ranked lattice and L is modular, then L is 2-distributive in András P. Huhn’s sense, and (3) if L is distributive, then RCSub(L) is a ranked lattice.

We prove that a tolerance relation of a lattice is a homomorphic image of a congruence relation.

The Swing Lemma, proved by G. Gratzer in 2015, describes how a congruence spreads from a prime interval to another in a slim (having no $$\mathsf {M}_{3}$$ sublattice), planar, semimodular lattice. We generalize the Swing Lemma to planar semimodular lattices.

A planar (upper) semimodular lattice $L$ is slim if the five-element nondistributive modular lattice $M_3$ does not occur among its sublattices. (Planar lattices are finite by definition.) Slim rectangular lattices as particular slim planar semimodular lattices were defined by G. Gr\"atzer and E. Knapp in 2007. In 2009, they also proved that the congruence lattices of slim planar semimodular lattices with at least three elements are the same as those of slim rectangular lattices. In order to provide an effective tool for studying these congruence lattices, we introduce the concept of lamps of slim rectangular lattices and prove several of their properties. Lamps and several tools based on them allow us to prove in a new and easy way that the congruence lattices of slim planar semimodular lattices satisfy the two previously known properties. Also, we use lamps to prove that these congruence lattices satisfy four new properties including the two-pendant four-crown property and the forbidden marriage property., Comment: 26 pages, 11 figures

Let $n>3$ be a natural number. By a 1975 result of H. Strietz, the lattice Part$(n)$ of all partitions of an $n$-element set has a four-element generating set. In 1983, L. Z\'adori gave a new proof of this fact with a particularly elegant construction. Based on his construction from 1983, the present paper gives a lower bound on the number $\nu(n)$ of four-element generating sets of Part$(n)$. We also present a computer assisted statistical approach to $\nu(n)$ for small values of $n$. In his 1983 paper, L. Z\'adori also proved that for $n\geq 7$, the lattice Part$(n)$ has a four element generating set that is not an antichain. He left the problem whether such a generating set for $n\in\{5,6\}$ exists open. Here we solve this problem in negative for $n=5$ and in affirmative for $n=6$. Finally, the main theorem asserts that the direct product of some powers of partition lattices is four-generated. In particular, by the first part of this theorem, Part$(n_1)\times$ Part$(n_2)$ is four-generated for any two distinct integers $n_1$ and $n_2$ that are at least 5. The second part of the theorem is technical but it has two corollaries that are easy to understand. Namely, the direct product Part$(n)$ $\times$ Part$(n+1)$ $\times\dots\times$ Part$(3n-14)$ is four-generated for each integer $n\geq 9$. Also, for every positive integer $u$, the $u$-th the direct power of the direct product Part$(n)$ $\times$ Part$(n+1)$ $\times\dots\times$ Part$(n+u-1)$ is four-generated for all but finitely many $n$. If we do not insist on too many direct factors, then the exponent can be quite large. For example, our theorem implies that the $10^{127}$-th direct power of Part$(1011)$ $\times$ Part$(1012)$ $\times \dots \times$ Part$(2020)$ is four-generated., Comment: 34 pages, 6 figures

Finite (upper) nearlattices are essentially the same mathematical entities as finite semilattices, finite commutative idempotent semigroups, finite join-enriched meet semilattices, and chopped lattices. We prove that if an $n$-element nearlattice has at least $83\cdot 2^{n-8}$ subnearlattices, then it has a planar Hasse diagram. For $n>8$, this result is sharp., Comment: 71 pages, 7 figures

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., Comment: 15 pages, 2 figures

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., Comment: 28 pages, 7 figures

We describe the absolute retracts for the following classes of finite lattices: (1) slim semimodular lattices, (2) finite distributive lattices, and for each positive integer $n$, (3) at most $n$-dimensional finite distributive lattices. Although the singleton lattice is the only absolute retract for the first class, this result has paved the way to some other classes. For the second class, we prove that the absolute retracts are exactly the finite boolean lattices; this generalizes a 1979 result of J. Schmid. For the third class, the absolute retracts are the finite boolean lattices of dimension at most $n$ and the direct products of $n$ nontrivial finite chains. Also, we point out that in each of these classes, the algebraically closed lattices and the strongly algebraically closed lattices are the same as the absolute retracts. Slim (and necessarily planar) semimodular lattices were introduced by G. Gr\"atzer and E. Knapp in 2007, and they have been intensively studied since then. Algebraically closed and strongly algebraically closed lattices have been investigated by J. Schmid and, in several papers, by A. Molkhasi., Comment: 22 pages, 2 figures

A lattice is $(1+1+2)$-generated if it has a four-element generating set such that exactly two of the four generators are comparable. We prove that the lattice Quo$(n)$ of all quasiorders (also known as preorders) of an $n$-element set is $(1+1+2)$-generated for $n=3$ (trivially), $n=6$ (when Quo(6) consists of $209\,527$ elements), n=11, and for every natural number $n\geq 13$. In 2017, the second author and J. Kulin proved that Quo$(n)$ is $(1+1+2)$-generated if either $n$ is odd and at least $13$ or $n$ is even and at least $56$. Compared to the 2017 result, this paper presents twenty-four new numbers $n$ such that Quo$(n)$ is $(1+1+2)$-generated. Except for Quo(6), an extension of Z\'adori's method is used., Comment: 14 pages, 4 figures

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 counterpart of this fact for all bipartite graphs in the class of ALL graphs is a well-known consequence of the compactness theorem.) Also, to exemplify that our method is applicable in various fields of mathematics, we prove that neither finite simple groups, nor the ordered sets of join-irreducible congruences of slim semimodular lattices can be described by finitely many axioms in the class of FINITE structures. Since a 2007 result of G. Gr\"atzer and E. Knapp, slim semimodular lattices have constituted the most intensively studied part of lattice theory and they have already led to results even in group theory and geometry. In addition to the non-axiomatizability results mentioned above, we present a new property, called Decomposable Cyclic Elements Property, of the congruence lattices of slim semimodular lattices., Comment: 19 pages, 4 figures

For an integer $n\geq 5$, H. Strietz (1975) and L. Z\'adori (1986) proved that the lattice Part$(n)$ of all partitions of $\{1,2,\dots,n\}$ is four-generated. Developing L. Z\'adori's particularly elegant construction further, we prove that even the $k$-th direct power Part$(n)^k$ of Part$(n)$ is four-generated for many but only finitely many exponents $k$. E.g., Part$(n)^k$ is four-generated for every $k\leq 3\cdot 10^{89}$, and it has a four element generating set that is not an antichain for every $k\leq 1.4\cdot 10^{34}$. In connection with these results, we outline a protocol how to use these lattices in authentication and secret key cryptography., Comment: 20 pages, 5 figures

For bounded lattices L1 and L2, let $${f\colon L_1 \to L_2}$$ be a lattice homomorphism. Then the map $${{\rm Princ}(f)\colon \rm {Princ}(\it L_1) \to {\rm Princ}(\it L_2)}$$ , defined by $${{\rm con}(x,y) \mapsto {\rm con}(f(x),f(y))}$$ , is a 0-preserving isotone map from the bounded ordered set Princ(L1) of principal congruences of L1 to that of L2. We prove that every 0-preserving isotone map between two bounded ordered sets can be represented in this way. Our result generalizes a 2016 result of G. Gratzer from $${\{0,1}\}$$ -preserving isotone maps to 0-preserving isotone maps.

As the main achievement of the paper, we construct a three-generated, 2-distributive, atomless lattice that is not finitely presented. Also, the paper contains the following three observations. First, every coatomless three-generated lattice has at least one atom. Second, we give some sufficient conditions implying that a three-generated lattice has at most three atoms. Third, we present a three-generated meet-distributive lattice with four atoms., Comment: 12 pages, 3 figures

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., Comment: 24 pages, 4 figures (plus a logo). In the earlier version, the main result stated less then promised in the abstract. This error and some small imperfections have been corrected

Let $L$ be a lattice of finite length and let $d$ denote the minimum path length metric on the covering graph of $L$. For any $\xi=(x_1,\dots,x_k)\in L^k$, an element $y$ belonging to $L$ is called a median of $\xi$ if the sum $d(y,x_1)+\cdots+d(y,x_k)$ is minimum. The lattice $L$ satisfies the $c_1$-median property if, for any $\xi=(x_1,\dots,x_k)\in L^k$ and for any median $y$ of $\xi$, $y\leq x_1\vee\dots\vee x_k$. Our main theorem asserts that if $L$ is an upper semimodular lattice of finite length and the breadth of $L$ is less than or equal to $2$, then $L$ satisfies the $c_1$-median property. Also, we give a construction that yields semimodular lattices, and we use a particular case of this construction to prove that our theorem is sharp in the sense that $2$ cannot be replaced by $3$., Comment: 12 pages, 1 figure

Let $L$ be a finite $n$-element lattice. We prove that if $L$ has at least $83\cdot 2^{n-8}$ sublattices, then $L$ is planar. For $n>8$, this result is sharp since there is a non-planar lattice with exactly $83\cdot 2^{n-8}-1$ sublattices., Comment: 100 pages, 5 figures

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($\omega$)=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($\omega)$ 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$(\omega)$ 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$(\omega)$ into FL(3) in a TOTALLY SYMMETRIC way. Our main result determines all pairs $(\kappa,\lambda)$ of cardinals greater than 2 such that FL$(\kappa)$ is embeddable into FL$(\lambda)$ in a totally symmetric way. Also, we relax the stipulations on $S\cong$FL$\kappa$ by requiring only that $S$ is closed with respect to the automorphisms of FL$(\lambda)$, or $S$ is selfdually positioned and closed with respect to the automorphisms; we determine the corresponding pairs $(\kappa,\lambda)$ 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., Comment: 22 pages, 2 figures

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} $1-\epsilon$ 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 $1-\epsilon$. % 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 $\epsilon\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 $1-\epsilon$ 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 $\psi(E_1)$ is disjoint from $E_2$ for every non-degenerate affine transformation $\psi$., Comment: 18 pages, 6 figures

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., Comment: 18 pages, 15 figures

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., Comment: 10 pages, 2 figures

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-T\'oth. A theorem of L. Fejes-T\'oth 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-T\'oth'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-T\'oth'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., Comment: 14 pages, 9 figures

We prove that every finite lattice L can be embedded in a three-generated finite lattice K. We also prove that every algebraic lattice with accessible cardinality is a complete sublattice of an appropriate algebraic lattice K such that K is completely generated by three elements. Note that ZFC has a model in which all cardinal numbers are accessible. Our results strengthen P. Crawley and R. A. Dean's 1959 results by adding finiteness, algebraicity, and completeness., Comment: 11 pages, 3 figures

Let L be a lattice of finite length, ξ = (x1,…, xk)∈Lk, and y ∈ L. The remotenessr(y, ξ) of y from ξ is d(y, x1)+⋯+d(y, xk), where d stands for the minimum path length distance in the covering graph of L. Assume, in addition, that L is a graded planar lattice. We prove that whenever r(y, ξ) ≤ r(z, ξ) for all z ∈ L, then y ≤ x1∨⋯∨xk. In other words, L satisfies the so-called c1-median property. © 2015 Springer Science+Business Media Dordrecht

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" $\delta$ from the first point to the last one is at most $2\pi$. We prove that for $n\geq 3$, the $n$-fan cannot be constructed with straightedge and compass in general from its central angle $\delta$ and its central distances, which are the distances of the edges from the center of the circle. Also, we prove that for each fixed $\delta$ in the interval $(0, 2\pi]$ and for every $n\geq 5$, there exists a concrete $n$-fan with central angle $\delta$ that is not constructible from its central distances and $\delta$. The present paper generalizes some earlier results published by the second author and \'A. Kunos on the particular cases $\delta=2\pi$ and $\delta=\pi$., Comment: 12 pages, 1 figure

Let $$\mathbf {Lat}^{\mathrm{sd}}_{5}$$ and $$\mathbf {Pos}_{01}^{{\scriptscriptstyle \mathrm{{+}}}}$$ denote the category of selfdual bounded lattices of length 5 with $$\{0,1\}$$ -preserving lattice homomorphisms and that of bounded ordered sets with $$\{0,1\}$$ -preserving isotone maps, respectively. For an object L in $$\mathbf {Lat}^{\mathrm{sd}}_{5}$$ , the ordered set of principal congruences of the lattice L is denoted by $$\mathrm{Princ}(L)$$ . By means of congruence generation, $$\mathrm{Princ}:\mathbf {Lat}^{\mathrm{sd}}_{5}\rightarrow \mathbf {Pos}_{01}^{{\scriptscriptstyle \mathrm{{+}}}}$$ is a functor. We prove that if $$\mathbf {A}$$ is a small subcategory of $$\mathbf {Pos}_{01}^{{\scriptscriptstyle \mathrm{{+}}}}$$ such that every morphism of $$\mathbf {A}$$ is a monomorphism, understood in $$\mathbf {A}$$ , then $$\mathbf {A}$$ is the $$\mathrm{Princ}$$ -image of an appropriate subcategory of $$\mathbf {Lat}^{\mathrm{sd}}_{5}$$ . This result extends G. Gratzer’s earlier theorems where $$\mathbf {A}$$ consisted of one or two objects and at most one non-identity morphism, and the author’s earlier result where all morphisms of $$\mathbf {A}$$ were 0-separating and no hom-set had more the two morphisms. Furthermore, as an auxiliary tool, we derive some families of maps, also known as functions, from injective maps and surjective maps; this can be useful in various fields of mathematics, not only in lattice theory. Namely, for every small concrete category $$\mathbf {A}$$ , we define a functor $${F_{{\scriptscriptstyle \mathrm{{com}}}}}$$ , called cometic functor, from $$\mathbf {A}$$ to the category $$\mathbf Set $$ of sets and a natural transformation $${{\varvec{\pi }}^{{\scriptscriptstyle \mathrm{{com}}}}}$$ , called cometic projection, from $${F_{{\scriptscriptstyle \mathrm{{com}}}}}$$ to the forgetful functor of $$\mathbf {A}$$ into $$\mathbf Set $$ such that the $${F_{{\scriptscriptstyle \mathrm{{com}}}}}$$ -image of every monomorphism of $$\mathbf {A}$$ is an injective map and the components of $${{\varvec{\pi }}^{{\scriptscriptstyle \mathrm{{com}}}}}$$ are surjective maps.

Let $\lambda$ and $\kappa$ be cardinal numbers such that $\kappa$ is infinite and either $2\leq \lambda\leq \kappa$, or $\lambda=2^\kappa$. We prove that there exists a lattice $L$ with exactly $\lambda$ many congruences, $2^\kappa$ many ideals, but only $\kappa$ many filters. Furthermore, if $\lambda\geq 2$ 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 $\lambda=2$. Related to some earlier results of George Gr\"atzer 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 $\kappa$ is an infinite cardinal such that $\kappa\geq |P|$ and $\kappa\geq |G|$, then there exists a lattice $L$ of cardinality $\kappa$ 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 $2^\kappa$ many ideals, but (iv) $L$ has only $\kappa$ many filters., Comment: 13 pages, 3 figures

Motivated by a recent paper of G. Gr\"atzer, 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. Gr\"atzer, 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. Gr\"atzer 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., Comment: 20 pages, 8 figures

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. Gr\"atzer. 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., Comment: 15 pages, 2 figures (Note that Section 5 and Proposition 5 are new.)

Let F be a finite set of circles in the plane. We point out that the usual convex closure restricted to F yields a convex geometry, that is, a combinatorial structure introduced by P. H Edelman in 1980 under the name "anti-exchange closure system". We prove that if the circles are collinear and they are arranged in a "concave way", then they determine a convex geometry of convex dimension at most 2, and each finite convex geometry of convex dimension at most 2 can be represented this way. The proof uses some recent results from Lattice Theory, and some of the auxiliary statements on lattices or convex geometries could be of separate interest. The paper is concluded with some open problems., Comment: 22 pages, 7 figures

With the help of our new tools in the title, we give an efficient representation of the congruence lattice of a slim rectangular lattice by an easy-to-visualize quasiordering on the set of its meet-irreducible elements or, equivalently, on the set of its trajectories.

Join-distributive lattices are finite, meet-semidistributive, and semimodular lattices. They are the same as Dilworth's lattices in 1940, and many alternative definitions and equivalent concepts have been discovered or rediscovered since then. Let L be a join-distributive lattice of length n and let k denote the width of the set of join-irreducible elements of L. A result of P.H. Edelman and R.E. Jamison, translated from Combinatorics to Lattice Theory, says that L can be described by k-1 permutations acting on the set {1,...,n}. We prove a similar result within Lattice Theory: there exist k-1 permutations acting on {1,...,n} such that the elements of L are coordinatized by k-tuples over {0,...,n}, and the permutations determine which k-tuples are allowed. Since the concept of join-distributive lattices is equivalent to that of antimatroids and convex geometries, our result offers a coordinatization for these combinatorial structures.

A subset $X$ of a finite lattice $L$ is CD-independent if the meet of any two incomparable elements of $X$ equals 0. In 2009, Cz\'edli, Hartmann and Schmidt proved that any two maximal CD-independent subsets of a finite distributive lattice have the same number of elements. In this paper, we prove that if $L$ is a finite meet-distributive lattice, then the size of every CD-independent subset of $L$ is at most the number of atoms of $L$ plus the length of $L$. If, in addition, there is no three-element antichain of meet-irreducible elements, then we give a recursive description of maximal CD-independent subsets. Finally, to give an application of CD-independent subsets, we give a new approach to count islands on a rectangular board., Comment: 14 pages, 4 figures

In 2009, G. Gr\"atzer and E. Knapp proved that every planar semimodular lattice has a rectangular extension. We prove that, under reasonable additional conditions, this extension is unique. This theorem naturally leads to a hierarchy of special diagrams of planar semimodular lattices. Besides that these diagrams are unique in a strong sense, we explore many of their further properties. Finally, we demonstrate the power of our new diagrams in two ways. First, we prove a simplified version of our earlier Trajectory Coloring Theorem, which describes the inclusion con(p)\supseteq\con(q) for prime intervals p and q in slim rectangular lattices. Second, we prove G. Gr\"atzer's Swing Lemma for the same lattices, which describes the same inclusion more simply., Comment: 48 pages, 14 figures

By H. Strietz, 1975, and G. Cz\'edli, 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. Cz\'edli, 1996, G. Tak\'ach, 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., Comment: 8 pages, 3 figures

Let L denote the variety of lattices. In 1982, the second author proved that L is strongly tolerance factorable, that is, the members of L have quotients in L modulo tolerances, although L has proper tolerances. We did not know any other nontrivial example of a strongly tolerance factorable variety. Now we prove that this property is preserved by forming independent joins (also called products) of varieties. This enables us to present infinitely many {strongly} tolerance factorable varieties with proper tolerances. Extending a recent result of G.\ Cz\'edli and G.\ Gr\"atzer, we show that if V is a strongly tolerance factorable variety, then the tolerances of V are exactly the homomorphic images of congruences of algebras in V. Our observation that (strong) tolerance factorability is not necessarily preserved when passing from a variety to an equivalent one leads to an open problem., Comment: 10 pages

A recent result of G. Cz\'edli and E.\,T. Schmidt gives a construction of slim (planar) semimodular lattices from planar distributive lattices by adding elements, adding "forks". We give a construction that accomplishes the same by deleting elements, by "resections"., Comment: 12 pages, 6 figures

Let H ? and K ? be finite composition series of length h in a group G . The intersections of their members form a lattice CSL ( H ? , K ? ) under set inclusion. Our main result determines the number N ( h ) of (isomorphism classes) of these lattices recursively. We also show that this number is asymptotically h ! / 2 . If the members of H ? and K ? are considered constants, then there are exactly h ! such lattices.Based on recent results of Czedli and Schmidt, first we reduce the problem to lattice theory, concluding that the duals of the lattices CSL ( H ? , K ? ) are exactly the so-called slim semimodular lattices, which can be described by permutations. Hence the results on h ! and h ! / 2 follow by simple combinatorial considerations. The combinatorial argument proving the main result is based on Czedli's earlier description of indecomposable slim semimodular lattices by matrices.

Rectangular lattices are special planar semimodular lattices introduced by G. Gratzer and E. Knapp in Acta Sci Math 75:29–48, 2009. A patch lattice is a rectangular lattice whose weak corners are coatoms. As a variant of gluing, we introduce the concept of a patchwork system. We prove that every glued sum indecomposable, planar, semimodular lattice is a patchwork of its maximal patch lattice intervals. For a planar modular lattice, our patchwork system is the same as the S-glued system introduced by C. Herrmann in Math Z 130:255–274, 1973. Among planar semimodular lattices, patch lattices are characterized as the patchwork-irreducible ones. They are also characterized as the indecomposable ones with respect to gluing over chains; this gives another structure theorem.

For each of the relations “less than or equal to”, “less than”, “covered by”, and “covered by or equal to”, we characterize finite orders (also called posets) with the property that the pair of Galois closure operators induced by the relation in question coincides with the pair of closure operators introduced and applied in our previous paper in 2007. We also consider the “less than or equal to” relation between the set of join-irreducible elements and the set of meet-irreducible elements, and we show that the above-mentioned pairs of closure operators coincide for finite modular lattices.

For subnormal subgroups \({A{\vartriangleleft}B}\) and \({C{\vartriangleleft}D}\) of a given group G, the factor B/A will be called subnormally down-and-up projective to D/C if there are subnormal subgroups \({X{\vartriangleleft}Y}\) such that \({AY = B, A \cap Y = X, CY = D}\) , and \({C \cap Y = X}\) . Clearly, \({B/A \cong D/C}\) in this case. As G. Gratzer and J. B. Nation have recently pointed out, the standard proof of the classical Jordan-Holder theorem yields somewhat more than is widely known; namely, the factors of any two given composition series are the same up to subnormal down-and-up projectivity and a permutation. We prove the uniqueness of this permutation.