Related papers: Sublattices of complete lattices with continuity c…
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…
A lattice in Euclidean $d$-space is called well-rounded if it contains $d$ linearly independent vectors of minimal length. This class of lattices is important for various questions, including sphere packing or homology computations. The…
Lattices induced by coverings arise naturally in matroid theory and combinatorial optimization, providing a structured framework for analyzing relationships between independent sets and closures. In this paper, we explore the structural…
We prove that for any distributive join-semilattice S, there are a meet-semilattice P with zero and a map f:PxP-->S such that f(x,z)<=f(x,y)vf(y,z) and x<=y implies that f(x,y)=0, for all x,y,z in P, together with the following conditions:…
We investigate the representation of lattices as sublattices of the lattice of all convex subsets (intervals) of a linearly ordered set $(X,\le)$. We introduce the purely lattice-theoretic notion of a \textit{loc-lattice} and prove that…
We characterize factor congruences in semilattices by using generalized notions of order ideal and of direct sum of ideals. When the semilattice has a minimum (maximum) element, these generalized ideals turn into ordinary (dual) ideals.
For a closure space (P,f) with f(\emptyset)=\emptyset, the closures of open subsets of P, called the regular closed subsets, form an ortholattice Reg(P,f), extending the poset Clop(P,f) of all clopen subsets. If (P,f) is a finite convex…
Dilworth's theorem. Every finite distributive lattice $D$ can be represented as the congruence lattice of a finite lattice $L$. We want: Every finite distributive lattice $D$ can be represented as the congruence lattice of a nice finite…
We present two results related to an edge-isoperimetric question for Cayley graphs on the integer lattice asked by Ben Barber and Joshua Erde [Isoperimetry of Integer Lattices, Discrete Analysis 7 (2018)]. For any (undirected) graph $G$,…
For a finite distributive lattice $D$, let us call $Q \subseteq D$ \emph{principal congruence representable}, if there is a finite lattice $L$ such that the congruence lattice of $L$ is isomorphic to $D$ and the principal congruences of $L$…
The Johnson-Lindenstrauss (JL) lemma allows subsets of a high-dimensional space to be embedded into a lower-dimensional space while approximately preserving all pairwise Euclidean distances. This important result has inspired an extensive…
We extend the theory of atomized semilattices to the infinite setting. We show that it is well-defined and that every semilattice is atomizable. We also study atom redundancy, focusing on complete and finitely generated semilattices and…
In the article a classification method for nonlinear integrable equations with three independent variables is discussed based on the notion of the integrable reductions. We call the equation integrable if it admits a large class of…
The maximal index of a Euclidean lattice L of dimension n is the maximal index of the sub-lattices of L spanned by n independent minimal vectors of L. In this paper, we prove that a perfect lattice of maximal index two not provided by a…
Let $S$ be a non-empty, closed subspace of a locally compact group $G$ that is a subsemigroup of $G$. Suppose that $X, Y$, and $Z$ are Banach lattices that are vector sublattices of the order dual $\mathrm{C}_{\mathrm{c}}(S,\mathbb R)^\sim$…
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.…
If a partition of a lattice in R^d is selfsimilar, it is called lattice substitution system (LSS). Such sets represent nonperiodic, but highly ordered structures. An important property of such structures is, whether they are model sets or…
We investigate theoretically the topological properties of dimerized quasi-one-dimensional (1D) lattice comprising of multi legs $(L)$ as well as multi sublattices $(R)$. The system has main and subsidiary exchange symmetries. In the basis…
We prove that an infinite (bounded) involution lattice and even pseudo--Kleene algebra can have any number of congruences between $2$ and its number of elements or equalling its number of subsets, regardless of whether it has as many ideals…
The classification of lattice equations that are integrable in the sense of higher-dimensional consistency is extended by allowing directed edges. We find two cases that are not transformable via the 'admissible transformations' to the…