Related papers: Planar, infinite, semidistributive lattices

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…

We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice…

For a finite lattice L, the congruence lattice Con L of L can be easily computed from the partially ordered set J(L) of join-irreducible elements of L and the join-dependency relation D\_L on J(L). We establish a similar version of this…

A lattice $L$ is said lowly finite if the set $[\mathsf{0},a]$ is finite for every element $a$ of $L$. We mainly aim to provide a complete proof that, if $M$ is a subset of a complete lowly finite distributive lattice $L$ containing its…

For a presentation $\mathcal{A}$ of a transversal matroid $M$, we study the set $T_{\mathcal{A}}$ of single-element transversal extensions of $M$ that have presentations that extend $\mathcal{A}$; we order these extensions by the weak…

We prove that every distributive algebraic lattice with at most $\aleph\_1$ compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The $\aleph\_1$ bound is optimal, as…

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…

Given a join semilattice $S$ with a minimum $\hat{0}$, the quarks (also called atoms in order theory) are the elements that cover $\hat{0}$, and for each $x \in S \setminus \{\hat{0}\}$ a factorization (into quarks) of $x$ is a minimal set…

We show that for every quasivariety K of structures (where both functions and relations are allowed) there is a semilattice S with operators such that the lattice of quasi-equational theories of K (the dual of the lattice of…

In this article we introduce the study of the number of pairs of non-comparable elements in a distributive lattice $\L$. We give several tight lower and upper bounds for the number and give as an application the lattices precisely for which…

We prove a general categorical theorem that enables us to state that under certain conditions, the range of a functor is large. As an application, we prove various results of which the following is a prototype: If every diagram, indexed by…

We show that if $\mathcal{F}$ is an algebraically integrable foliation on a $\mathbb{Q}$-factorial normal projective variety $X$, $ A, B \geq 0$ are $\mathbb{Q}$-divisors on $X$ with $A$ ample such that $(\mathcal{F}, B)$ is foliated dlt…

Various embedding problems of lattices into complete lattices are solved. We prove that for any join-semilattice S with the minimal join-cover refinement property, the ideal lattice IdS of S is both algebraic and dually algebraic.…

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 $127\cdot 2^{n-8}-1$…

We prove that the problems of representing a finite ordered complemented semigroup or finite lattice-ordered semigroup as an algebra of binary relations over a finite set are undecidable. In the case that complementation is taken with…

Unitally nondistributive quantales are unital quantales such that the unit is approximable by the totally below relation and does not meet-distribute over arbitrary joins. It is shown that the underlying nondistributive complete lattice…

In a recent paper, the authors have proved that for lattices A and B with zero, the isomorphism $Conc(A \otimes B)\cong Conc A \otimes Conc B$, holds, provided that the tensor product satisfies a very natural condition (of being…

For a finite lattice L, let EL denote the reflexive and transitive closure of the join-dependency relation on L, defined on the set J(L) of all join-irreducible elements of L. We characterize the relations of the form EL, as follows:…

Let $L$ be a planar semimodular lattice. We call $L$ \emph{slim}, if it has no $\mthree$ sublattice. Let us define an \emph{SPS lattice} as a slim, planar, semimodular lattice $L$. In 2016, I proved a property of congruences of SPS lattices…

A planar semimodular lattice $L$ is \emph{slim} if $\mathbf{M}_3$ is not a sublattice of $L$. In a recent paper, G. Cz\'edli introduced a very powerful diagram type for slim, planar, semimodular lattices. This short note proves the…