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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…
This note reformulates certain classical combinatorial duality theorems in the context of order lattices. For source-target networks, we generalize bottleneck path-cut and flow-cut duality results to edges with capacities in a distributive…
In the latest developments in the theory of skew lattices, distributivity has been one of the main topics of study. The largest classes of examples of such algebras are distributive. Unlike what happens in lattices, the properties of…
We introduce the Birkhoff completion as the smallest distributive lattice in which a given finite lattice can be embedded as semi-lattice. We discuss its relationship to implicational theories, in particular to R. Wille's…
The congruence lattices of all algebras defined on a fixed finite set $A$ ordered by inclusion form a finite atomistic lattice $\mathcal E$. We describe the atoms and coatoms. Each meet-irreducible element of $\mathcal E$ being determined…
In 1876 H. J. S. Smith defined an LCM matrix as follows: let S = {x_1, x_2, ..., x_n} be a set of positive integers. The LCM matrix [S] is the n $\times$ n matrix with lcm(x_i , x_j) as its ij entry. During the last 30 years singularity of…
We consider hamiltonian systems with spatially varying effective mass and slowly varying local potential in d dimensions. The Slater sum is defined as the diagonal element of the Bloch propagator. We derive a gradient expansion of the…
A distributive lattice $L$ with minimum element $0$ is called decomposable if $a$ and $b$ are not comparable elements in $L$ then there exist $\overline{a},\overline{b}\in L$ such that $a=\overline{a}\vee(a\wedge b),…
We show how to derive Catterall's supersymmetric lattice gauge theories directly from the general principle of orbifolding followed by a variant of the usual deconstruction. These theories are forced to be complexified due to a clash…
In 2012 the second author obtained a description of the lattice of subgroupsof a Chevalley group $G(\Phi,A)$, containing the elementary subgroup $E(\Phi,K)$ over a subring $K\subseteq A$ provided $\Phi=B_n,$ $C_n$ or $F_4$, $n\ge2$, and $2$…
In 1945-46, C. L. Siegel proved that an $n$-dimensional lattice $\Lambda $ of determinant ${\rm det}(\Lambda )$ has at most $m^{n^2}$ different sublattices of determinant $m\cdot {\rm det}(\Lambda )$. In 1997, the exact number of the…
It is known from Grzegorczyk's paper \cite{grze-1951} that the lattice of real semi-algebraic closed subsets of ${\mathbb R}^n$ is undecidable for every integer $n\geq 2$. More generally, if $X$ is any definable set over a real or…
A simple but elegant result of Rival states that every sublattice $L$ of a finite distributive lattice $\mathcal{P}$ can be constructed from $\mathcal{P}$ by removing a particular family $\mathcal{I}_L$ of its irreducible intervals.…
Birkhoff's representation theorem for finite distributive lattices states that any finite distributive lattice is isomorphic to the lattice of order ideals (lower sets) of the partial order of the join-irreducible elements of the lattice.…
Covering is a common type of data structure and covering-based rough set theory is an efficient tool to process this data. Lattice is an important algebraic structure and used extensively in investigating some types of generalized rough…
We prove that every finite distributive lattice is isomorphic to a final segment of the d.c.e. Turing degrees (i.e., the degrees of differences of computably enumerable sets). As a corollary, we are able to infer the undecidability of the…
Let $L$ be any finite distributive lattice and $B$ be any boolean predicate defined on $L$ such that the set of elements satisfying $B$ is a sublattice of $L$. Consider any subset $M$ of $L$ of size $k$ of elements of $L$ that satisfy $B$.…
In 1980, U. Faigle introduced a sort of finite geometries on posets that are in bijective correspondence with finite semimodular lattices. His result has almost been forgotten in lattice theory. Here we simplify the axiomatization of these…
In this article, we continue the study of monadic distributive lattices (or m-lattices) which are a natural generalization of monadic Heyting algebras, introduced by Monteiro and Varsavsky and developed exhaustively by Bezhanishvili. First,…
A flat of a matroid is cyclic if it is a union of circuits; such flats form a lattice under inclusion and, up to isomorphism, all lattices can be obtained this way. A lattice is a Tr-lattice if all matroids whose lattices of cyclic flats…