The primitive cohomology lattice of a complete intersection

We describe the primitive cohomology lattice of a smooth even-dimensional complete intersection in projective space. To cite this article: A. Beauville, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On weighted lattice paths

A weighted lattice path from (1, 1) to (n, m) is a path consisting of unit vertical, horizontal, and diagonal steps of weight w. Let f(0), f(1), f(2), … be a nondecreasing sequence of positive integers; the path connecting the points of the set $\text{{(n, m) ¦ f(n ? 1) ? m ? f(n), n = 1, 2, …}}$ will be called the roof determined by f. We determine the number of weighted lattice paths from (1, 1) to (n + 1, f(n)) which do not cross the roof determined by f. We also determine the polynomials that must be placed in each cell below the roof such that if a 1 is placed in each cell whose lower left-hand corner is a point of the roof, every k × k square subarray comprised of adjacent rows and columns and containing at least one 1 will have determinant $\text{x(}\text{k}\text{2}\text{)}$.     

The convexity lattice of a poset

The authors investigate the lattice Co(P) of convex subsets of a general partially ordered set P. In particular, they determine the conditions under which Co(P) and Co(Q) are isomorphic; and give necessary and sufficient conditions on a lattice L so that L is isomorphic to Co(P) for some P.     

Method of solution to fuzzy relation equations in a complete Brouwerian lattice

Up to now, how to solve a fuzzy relation equation in a complete Brouwerian lattice is still an open problem as Di Nola et al. point out. To this problem, the key problem is whether there exists a minimal element in the solution set when a fuzzy relation equation is solvable. In this paper, we first show that there is a minimal element in the solution set of a fuzzy relation equation AX=b (where A=(a₁,a₂,…,a_n) and b are known, and X=(x₁,x₂,…,x_n)^T is unknown) when its solution set is nonempty, and b has an irredundant finite join-decomposition. Further, we give the method to solve AX=b in a complete Brouwerian lattice under the same conditions. Finally, a method to solve a more general fuzzy relation equation in a complete Brouwerian lattice when its solution set is nonempty is also given under similar conditions.     

Inversion of matrices over a pseudocomplemented lattice

We compute the greatest solutions of systems of linear equations over a lattice (P, ≤). We also present some applications of the results obtained to lattice matrix theory. Let (P, ≤) be a pseudocomplemented lattice with and and let A = ‖a _ij ‖ _n×n , where a _ij ∈ P for i, j = 1,..., n. Let A* = ‖a _ij ^′ ‖ _n×n and for i, j = 1,..., n, where a* is the pseudocomplement of a ∈ P in (P, ≤). A matrix A has a right inverse over (P, ≤) if and only if A · A* = E over (P, ≤). If A has a right inverse over (P, ≤), then A* is the greatest right inverse of A over (P, ≤). The matrix A has a right inverse over (P, ≤) if and only if A is a column orthogonal over (P, ≤). The matrix D = A · A* is the greatest diagonal such that A is a left divisor of D over (P, ≤). Invertible matrices over a distributive lattice (P, ≤) form the general linear group GL _n (P, ≤) under multiplication. Let (P, ≤) be a finite distributive lattice and let k be the number of components of the covering graph Γ(join(P,≤) − , ≤), where join(P, ≤) is the set of join irreducible elements of (P, ≤). Then GL _a (P, ≤) ≅ = S _n ^k . We give some further results concerning inversion of matrices over a pseudocomplemented lattice. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 139–154, 2005.     

Valuations on lattice polytopes

Let L be a lattice (that is, a Z-module of finite rank), and let L=P(L) denote the family of convex polytopes with vertices in L; here, convexity refers to the underlying rational vector space V=Q⊗L. In this paper it is shown that any valuation on L satisfies the inclusion-exclusion principle, in the strong sense that appropriate extension properties of the valuation hold. Indeed, the core result is that the class of a lattice polytope in the abstract group L=P(L) for valuations on L can be identified with its characteristic function in V. In fact, the same arguments are shown to apply to P(M), when M is a module of finite rank over an ordered ring, and more generally to appropriate families of (not necessarily bounded) polyhedra.     

On the lattice of stratified principal L-topologies

We investigate the lattice structure of the set of all stratified principal L-topologies on a given set X. It proves that the lattice of stratified principal L-topologies S p(X) has atoms and dual atoms if and only if L has atoms and dual atoms respectively. Moreover, it is complete and semi-complemented. We also discuss some other properties of the lattice.     

The rank of a distributive lattice

The rank of a partial ordering P is the maximum size of an irredundant family of linear extensions of P whose intersection is P. A simple relationship is established between the rank of a finite distributive lattice and its subset of join irreducible elements.     

The spectrum of a finite pseudocomplemented lattice

Let L be a finite pseudocomplemented lattice. Every interval [0, a] in L is pseudocomplemented, so by Glivenko’s theorem, the set S(a) of all pseudocomplements in [0, a] forms a boolean lattice. Let B _i denote the finite boolean lattice with i atoms. We describe all sequences (s ₀, s ₁, . . . , s _n ) of integers for which there exists a finite pseudocomplemented lattice L with s _i = |{ a ∈ L | S(a) ? B _i }|, for all i, and there is no a ∈ L with S(a) ? B _n+1. This result settles a problem raised by the first author in 1971.     

Laterally closed lattice homomorphisms

Let A and B be two Archimedean vector lattices and let be a lattice homomorphism. We call that T is laterally closed if T(D) is a maximal orthogonal system in the band generated by T(A) in B, for each maximal orthogonal system D of A. In this paper we prove that any laterally closed lattice homomorphism T of an Archimedean vector lattice A with universal completion Au into a universally complete vector lattice B can be extended to a lattice homomorphism of Au into B, which is an improvement of a result of M. Duhoux and M. Meyer [M. Duhoux and M. Meyer, Extended orthomorphisms and lateral completion of Archimedean Riesz spaces, Ann. Soc. Sci. Bruxelles 98 (1984) 3-18], who established it for the order continuous lattice homomorphism case. Moreover, if in addition Au and B are with point separating order duals ^′(Au) and B^′ respectively, then the laterally closedness property becomes a necessary and sufficient condition for any lattice homomorphism to have a similar extension to the whole Au. As an application, we give a new representation theorem for laterally closed d-algebras from which we infer the existence of d-algebra multiplications on the universal completions of d-algebras.     

The complete closure of a graph

We define the complete closure number cc(G) of a graph G of order n as the greatest integer k ≤ 2n ? 3 such that the kth Bondy-Chvátal closure Cl_k(G) is complete, and give some necessary or sufficient conditions for a graph to have cc(G) = k. Similarly, the complete stability cs(P) of a property P defined on all the graphs of order n is the smallest integer k such that if Cl_k(G) is complete then G satisfies P. For some properties P, we compare cs(P) with the classical stability s(P) of P and show that cs(P) may be far smaller than s(P). © 1993 John Wiley & Sons, Inc.     

Algebraic-polynomial approximation of functions satisfying a Lipschitz condition

For functionsf(x) ε KH^(α) [satisfying the Lipschitz condition of order α (0 < α < 1) with constant K on [?1, 1], the existence is proved of a sequence P_n (f; x) of algebraic polynomials of degree n = 1, 2,..., such that $$|f(x) - P_{n - 1} (f;x)| \leqslant \mathop {\sup }\limits_{f \in KH^{(\alpha )} } E_n (f)[(1 - x^2 )^{a/2} + o(1)],$$ when n → ∞, uniformly for x ε [?1, 1], where E_n(f) is the best approximation off(x) by polynomials of degree not higher than n.     

A selfdual embedding of the free lattice over countably many generators into the three-generated one

By a 1941 result of Whitman, the free lattice FL(3) = FL(x, y, z) includes a sublattice FL(ω) freely generated by infinitely many elements. Let δ denote the unique dual automorphism of FL(x, y, z) that acts identically on the set {x, y, z} of generators. We prove that FL(x, y, z) has a sublattice S isomorphic to FL(ω) such that δ(S) =  S.     

Representing an isotone map between two bounded ordered sets by principal lattice congruences

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. Grätzer from \({\{0,1}\}\)-preserving isotone maps to 0-preserving isotone maps.     

The lattice of convex subsemilattices of a semilattice

The convex subsemilattices of a semilattice E form a lattice Co(E) in the natural way. The purpose of this paper is to study how the properties of this lattice relate to the semilattice itself. For instance, lower semimodularity of the lattice is equivalent, along with various properties, to the semilattice being a tree. When E has more than two elements the lattice does, however, fail many common lattice-theoretic tests. It turns out that it is more fruitful to describe those semilattices E for which every “atomically generated” filter of Co(E) satisfies certain lattice-theoretic properties.     

Sums of sets of lattice points and unimodular coverings of polytopes

If P is a lattice polytope (that is, the convex hull of a finite set of lattice points in \({\mathbf{R}^n}\)), then every sum of h lattice points in P is a lattice point in the h-fold sumset hP. However, a lattice point in the h-fold sumset hP is not necessarily the sum of h lattice points in P. It is proved that if the polytope P is a union of unimodular simplices, then every lattice point in the h-fold sumset hP is the sum of h lattice points in P.     

Principle of inclusion-exclusion on semilattices

We introduce an enumeration theorem under lattice action. Let L be a finite semilattice and Ω be a nonempty set. Let $\text{f: L → P(Ω)}$ be a map satisfying f(x ? y) ? f(x) ∩ f(y), where ? and $\text{P}$(Ω) mean “join” and the power set of Ω, respectively. Then

$\text{m}\text{∪}\text{x?L}\text{?(x)}\text{=}\text{Σ}\text{c?C}\text{(?1)}{}^{\mathrm{l(c)}}\text{m}\text{∩}\text{x?c}\text{?(x)}$

, where C is the set of all chains in L and l(c) denotes the length of a chain c. Also the theorem can be dualized. Furthermore, we describe two applications of the theorem to a Boolean lattice of sets and a partition lattice of a set.     

Topologies on abelian lattice ordered groups induced by a positive filter and completeness

We consider topologies on an abelian lattice ordered group that are determined by the absolute value and a positive filter. We show that the topological completions of these objects are also determined by the absolute value and a positive filter. We investigate the connection between the topological completion of such objects and the Dedekind–MacNeille completion of the underlying lattice ordered group. We consider the preservation of completeness for such topologies with respect to homomorphisms of lattice ordered groups. Finally, we show that topologies defined in terms of absolute value and a positive filter on the space C(X) of all real-valued continuous functions defined on a completely regular topological space X are always complete.     

Approximate and complete controllability of nonlinear systems to a convex target set

The research deals with complete and approximate controllability of the system $\text{(1)}\text{dx}\text{dy}\text{= f(t, x, u)}$, without control restraints to an arbitrary convex target set. First, some characterizations of complete controllability, to the target of (1) and a special case of (1) namely $\text{x}\text{?}\text{= A(t)x + k(t, u)}{}^7$ are given. As a consequence complete controllability is equivalent to null-controllability. Next certain equations are formulated. These are in the same spirit as J. P. Dauer's “A Controllability Technique for Nonlinear Systems” (J. Math. Anal. Appl. oo (1972), 442–451) and are utilized in the main contribution of the paper: Under certain convexity assumption, bounded perturbations of systems which are completely controllable to a fixed target G are completely controllable to G. Without the convexity assumption, but with perturbations satisfying a Lipschitz condition, approximate controllability to G of a perturbed system is equivalent to complete controllability to G of the unperturbed equation.     

Atoms, anti-atoms and complements in the lattice of quasi-uniformities

We describe the atoms of the complete lattice (q(X),⊆) of all quasi-uniformities on a given (nonempty) set X. We also characterize those anti-atoms of (q(X),⊆) that do not belong to the quasi-proximity class of the discrete uniformity on X. After presenting some further results on the adjacency relation in (q(X),⊆), we note that (q(X),⊆) is not complemented for infinite X and show how ideas about resolvability of (bi)topological spaces can be used to construct complements for some elements of (q(X),⊆).