Search

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 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 ${\rm J}({\rm Con} L)$ of join-irreducible congruences of $L$. We prove that if $1

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 \(22\) and \(6\,000\), 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.

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 the singleton poset, then \(S(U,n)\) 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.J. Stahl, and W.T.Jr. Trotter. We determine \(S(U,n)\) for all finite posets with 0 and 1, and we give reasonable estimates for the "\(V\)-shaped" 3-element poset and, mainly, for the 4-element poset with 0 and three maximal elements. For a lattice \(L\), let \(G_{\min}L\) 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 \(k\mapsto G_{\min}{D^k}\) is the left adjoint of the function \(n\mapsto S(U,n)\). This allows us to determine \(G_{\min}{D^k}\) in many cases. E.g., for a 5-element distributive lattice \(D\), \(G_{\min}{D^{2023}}=18\) if \(D\) is a chain and \(G_{\min}{D^{2023}}=15\) otherwise. The present paper, another recent paper, and a 2021 one indicate that large direct powers of small distributive lattices could be of interest in cryptography.

K. Adaricheva and M. Bolat have recently proved that if $\,\mathcal U_0$ and $\,\mathcal U_1$ 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 $\,\mathcal U_{1-k}$ is included in the convex hull of $\,\mathcal U_k\cup(\{A_0,A_1, A_2\}\setminus\{A_j\})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ for the more general case where $\,\mathcal U_0$ and $\,\mathcal U_1$ are compact sets in the plane such that $\,\mathcal U_1$ is obtained from $\,\mathcal U_0$ 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.

Abstract.  Let L be a bounded lattice. If for each a1 b1 ∈ L and a2 b2 ∈ L there is a lattice embedding ψ: [a1, b1] → [a2, b2] with ψ(a1) = a2 and ψ(b1) = b2, then we say that L is a quasifractal. If ψ can always be chosen to be an isomorphism or, equivalently, if L is isomorphic to each of its nontrivial intervals, then L will be called a fractal lattice. For a ring R with 1 let $${\mathcal{V}}(R)$$ denote the lattice variety generated by the submodule lattices of R-modules. Varieties of this kind are completely described in [16]. The prime field of characteristic p will be denoted by Fp. Let $$\mathcal{U}$$ be a lattice variety generated by a nondistributive modular quasifractal. The main theorem says that $$\mathcal{U}$$ is neither too small nor too large in the following sense: there is a unique $$p = p(\mathcal{U})$$, a prime number or zero, such that $${\mathcal{V}}(F_p) \subseteq \mathcal{U}$$ and for any n ≥ 3 and any nontrivial (normalized von Neumann) n-frame $$(\vec{a},\vec{c}) = (a_1, . . . , a_n, c_{12}, . . . , c_{1n})$$ of any lattice in $$\mathcal{U}$$, $$(\vec{a},\vec{c})$$ is of characteristic p. We do not know if $$\mathcal{U} = \mathcal V(F_p)$$ in general; however we point out that, for any ring R with 1, $$\mathcal V(R) \subseteq \mathcal{U}$$ implies $${\mathcal{V}}(R) = {\mathcal{V}}(F_p)$$. It will not be hard to show that $$\mathcal{U}$$ is Arguesian. The main theorem does have a content, for it has been shown in [2] that each of the $${\mathcal{V}}(F_p)$$ is generated by a single fractal lattice Lp; moreover we can stipulate either that Lp is a continuous geometry or that Lp is countable. The proof of the main theorem is based on the following result of the present paper: if $$(\vec{a},\vec{c})$$ is a nontrivial m-frame and $$(\vec{u},\vec{v})$$ is an n-frame of a modular lattice L with m, n ≥ 3 such that $$u_1 \vee \cdot \cdot \cdot \vee u_n = a_1$$ and $$u_1 \wedge u_2 = a_1 \wedge a_2$$, then these two frames have the same characteristic and, in addition, they determine a nontrivial mn-frame $$(\vec{b},\vec{d})$$ of the same characteristic in a canonical way, which we call the product frame. [ABSTRACT FROM AUTHOR]

Abstract.  We introduce definitions of semifractal, 0–1-fractal, quasifractal and fractal lattices. A variety generated by a fractal lattice is called fractal generated, with analogous terminology for the other variants. We show that a semifractal generated nondistributive lattice variety cannot be of residually finite length. This easily implies that there are exactly continuously many lattice varieties which are not semifractal generated. On the other hand, for each prime field F, the variety generated by all subspace lattices of vector spaces over F is shown to be fractal generated. These countably many varieties and the class $${\mathcal{D}}$$ of all distributive lattices are the only known fractal generated lattice varieties at present. Four distinct countable distributive fractal lattices are given each of which generates $${\mathcal{D}}$$. After showing that each lattice can be embedded in a quasifractal, continuously many quasifractals are given each of which has cardinality $${\aleph_{0}}$$ and generates the variety of all lattices. Semifractal considerations are applied to construct examples of convexities that include no minimal convexity, thus answering a question of Jakubík. (A convexity is a class of lattices closed under taking homomorphic images, convex sublattices and direct products, a notion due to Ervin Fried.) [ABSTRACT FROM AUTHOR]

I think it is hopeless to try to give a concise account of the over 300 publications of George Grätzer and E. Tamás Schmidt. There is a danger of getting lost in too many details (for instance, mentioning Grätzer [G35], where the concept of Mal’cev condition was introduced and named). Hence, I will be guided by the following five restrictions. (1) I will discuss only lattice theoretic results. (2) I will try to concentrate on their deepest results, (3) on results with the largest impact, and (4) on series of papers. (5) I will give preference to joint or at least partially joint works. Even then, I will be able to cover only a small part of their results satisfying these five criteria. [ABSTRACT FROM AUTHOR]

Abstract.  Weak congruence lattices and semidistributive congruence lattices are both recent topics in universal algebra. This motivates the main result of the present paper, which asserts that a finite group G is a Dedekind group if and only if the diagonal relation is a join-semidistributive element in the lattice of weak congruences of G. A variant in terms of subgroups rather than weak congruences is also given. It is pointed out that no similar result is valid for rings. An open problem and some results on the join-semidistributivity of weak congruence lattices are also included. [ABSTRACT FROM AUTHOR]

Abstract. For varieties, congruence modularity is equivalent to the tolerance intersection property, TIP in short. Based on TIP, it was proved in [5] that for an arbitrary lattice identity implying modularity (or at least congruence modularity) there exists a Mal’tsev condition such that the identity holds in congruence lattices of algebras of a variety if and only if the variety satisfies the corresponding Mal’tsev condition. However, the Mal’tsev condition constructed in [5] is not the simplest known one in general. Now we improve this result by constructing the best Mal’tsev condition and various related conditions. As an application, we give a particularly easy new proof of the result of Freese and Jónsson [11] stating that modular congruence varieties are Arguesian, and we strengthen this result by replacing “Arguesian” by “higher Arguesian” in the sense of Haiman [18]. We show that lattice terms for congruences of an arbitrary congruence modular variety can be computed in two steps: the first step mimics the use of congruence distributivity, while the second step corresponds to congruence permutability. Particular cases of this result were known; the present approach using TIP is even simpler than the proofs of the previous partial results. [ABSTRACT FROM AUTHOR]

For an arbitrary lattice identity implying modularity (or at least congruence modularity) a Mal'tsev condition is given such that the identity holds in congruence lattices of algebras of a variety if and only if the variety satisfies the corresponding Mal'tsev condition. [ABSTRACT FROM AUTHOR]

Gumm [6] used the Shifting Lemma with high success in congruence modular varieties. Later, some analogous diagrammatic statements, including the Triangular Scheme from [1] were also investigated. The present paper deals with the purely lattice theoretic underlying reason for the validity of these lemmas. The shift of a lattice identity, a special Horn sentence, is introduced. To any lattice identity ? and to any variable y occurring in ? we introduce a Horn sentence S(?, y). When S(?, y) happens to be equivalent to ?, we call it a shift of ?. When ? has a shift then it gives rise to diagrammatic statements resembling the Shifting Lemma and the Triangular Scheme. Some known lattice identities will be shown to have a shift while some others have no shift. [ABSTRACT FROM AUTHOR]