Sum Of Lattices Research Articles

Duals of Cesàro sequence vector lattices, Cesàro sums of Banach lattices, and their finite elements

In this paper, we study the ideals of finite elements in special vector lattices of real sequences, first in the duals of Cesàro sequence spaces {text {ces}}_p for pin {0}cup [1,infty ) and, second, after the Cesàro sum {text {ces}}_{p}{(mathfrak {X})} of a sequence of Banach spaces is introduced, where p=infty is also allowed, we characterize their duals and the finite elements in these sums if the summed up spaces are Banach lattices. This is done by means of a remarkable extension of the corresponding result for direct sums.

The pre-period of the glued sum of finite modular lattices

The pre-period of the glued sum of finite modular lattices

Exponential lower bounds of lattice counts by vertical sum and 2-sum

We consider the problem of finding lower bounds on the number of unlabeled n-element lattices in some lattice family. We show that if the family is closed under vertical sum, exponential lower bounds can be obtained from vertical sums of small lattices whose numbers are known. We demonstrate this approach by establishing that the number of modular lattices is at least 2.2726^n for n large enough. We also present an analogous method for finding lower bounds on the number of vertically indecomposable lattices in some family. For this purpose we define a new kind of sum, the vertical 2-sum, which combines lattices at two common elements. As an application we prove that the numbers of vertically indecomposable modular and semimodular lattices are at least 2.1562^n and 2.6797^n for n large enough.

Lattice-based sum of t-norms on bounded lattices

The concept of ordinal sums in the sense of Clifford have long been blamed for their limitations in constructing new t-norms including inability to cope with general bounded lattices. Motivated by this observation, and based on the lattice-based sum of lattices that has been recently introduced by El-Zekey et al., we propose a new sum-type construction of t-norms, called a lattice-based sum of t-norms, for building new t-norms on bounded lattices from given ones. The proposed sum is generalizing the well-known ordinal and horizontal sum constructions of t-norms by allowing for lattice ordered index sets. We demonstrate that, like the ordinal sum of t-norms, the lattice-based sum of t-norms can be generalized using as summands so-called t-subnorms, still leading to a t-norm. Subsequently, we apply the results for constructing several new families of t-norms and t-subnorms on bounded lattices. In the same spirit, by the duality, we will also introduce lattice-based sums of t-conorms and t-subconorms.

Lattice-based Sum Construction of Nullnorms on Bounded Lattices

Nullnorms on bounded lattices are generalizations of t-norms and t-conorms with a zero element laying anywhere in the underlying lattices. In this paper, new methods for constructing nullnorms on bounded lattices are proposed. The proposed construction methods are based on the lattice-based sum of lattices that has been recently introduced by El-Zekey et al. (see [9]), for building new (bounded) lattices from fixed ones indexed by a (finite) lattice-ordered index set. Subsequently, the new construction methods are applied for building several new families of nullnorms on bounded lattices. As a by-product, lattice-based sum constructions of t-norms and t-conorms on bounded lattices have been obtained. Furthermore, new idempotent nullnorms on bounded lattices have been also obtained.

Transcendental Hodge algebra

The transcendental Hodge lattice of a projective manifold M is the smallest Hodge substructure in pth cohomology which contains all holomorphic p-forms. We prove that the direct sum of all transcendental Hodge lattices has a natural algebraic structure, and compute this algebra explicitly for a hyperkahler manifold. As an application, we obtain a theorem about dimension of a compact torus T admitting a holomorphic symplectic embedding to a hyperkahler manifold M. If M is generic in a d-dimensional family of deformations, then $$\dim T\ge 2^{[(d+1)/2]}$$ .

Distributive lattice structure on the set of perfect matchings of carbon nanotubes

Carbon nanotubes are composed of carbon atoms linked in hexagonal shapes, with each carbon atom covalently bonded to three other carbon atoms. Carbon nanotubes have diameters as small as 1 nm and lengths up to several centimeters. Carbon nanotubes can be open-ended or closed-ended (fullerenes). Open-ended single-walled carbon nanotubes are also called tubulenes. The resonance graph R(T) of a tubulene T reflects interactions between Kekule structures—i.e. perfect matchings of T. With the orientation of edges the resonance digraph \(\overrightarrow{R}(T)\) of a tubulene is obtained. As the main result we show that \(\overrightarrow{R}(T)\) is isomorphic to the Hasse diagram of the direct sum of some distributive lattices. Similar results were proved in [10, 16], but one can not directly apply them to tubulenes. As a consequence of the main result it is proved that every connected component of R(T) is a median graph. Further we show that the block graph of every connected component H of the resonance graph of a tubulene is a path and that H contains at most two vertices of degree one.

ON INTERVAL-VALUED FUZZY LATTICES

We discuss the relationship between interval-valued fuzzy ideals and interval-valued fuzzy congruence on a distributive lattice L and show that for a generalized Boolean algebra the lattice of interval-valued fuzzy ideals is isomorphic to the lattice of interval-valued fuzzy congruences. Finally we consider the products of interval-valued fuzzy ideals and obtain a necessary and sufficient condition for an interval-valued fuzzy ideal on the direct sum of lattices to be representable as a direct sum of interval-valued fuzzy ideals on each lattice.

Projective Ordinal Sums of Lattices and Isotone Sections

This note gives a complete characterization of when the ordinal sum of two lattices (the lattice obtained by placing the second lattice on top of the first) is projective. This characterization applies not only to the class of all lattices, but to any variety of lattices, and in particular, to the class of distributive lattices. Lattices L with the property that every epimorphism onto L has an isotone section are also characterized.

Lattice-based sums

In this contribution, the well-known ordinal sum technique of posets is generalized by allowing for a lattice ordered index set instead of a linearly ordered index set, and we argue for the merits of this generalization. We will call such a proposed sum-type construction a lattice-based sum. Our new approach of lattice-based sum extends also the horizontal sum. We show that the lattice-based sum of posets is again a poset. Subsequently, we apply the results for constructing new lattices by investigating lattice-based sums when the summand posets are lattices. We show that under certain assumptions, the lattice-based sum of lattices will be a lattice.

Characterizations of Spectral Automorphisms and a Stone-Type Theorem in Orthomodular Lattices

The notion of spectral automorphism of an orthomodular lattice was introduced by Ivanov and Caragheorgheopol (Int. J. Theor. Phys. 49(12):3146–3152, 2010) to create an analogue of the Hilbert space spectral theory in the abstract framework of orthomodular lattices. We develop the theory of spectral automorphisms finding previously missing characterizations of spectral automorphisms, discussing several examples and the possibility to construct such automorphisms in direct products or horizontal sums of lattices. A factorization of the spectrum of a spectral automorphism is found. The last part of the paper addresses the problem of the unitary time evolution of a system from the point of view of the spectral automorphisms theory. An analogue of the Stone theorem concerning strongly continuous one-parameter unitary groups is given.

Direct Sum of Distributive Lattices on the Perfect Matchings of a Plane Bipartite Graph

Let G be a plane bipartite graph and \({\cal M}(G)\) the set of perfect matchings of G. A property that the Z-transformation digraph of perfect matchings of G is acyclic implies a partially ordered relation on \({\cal M}(G)\). It was shown that \({\cal M}(G)\) is a distributive lattice if G is (weakly) elementary. Based on the unit decomposition of alternating cycle systems, in this article we show that the poset \({\cal M}(G)\) is direct sum of finite distributive lattices if G is non-weakly elementary; Further, if G is elementary, then the height of distributive lattice \({\cal M}(G)\) equals the diameter of Z-transformation graph, and both quantities have a sharp upper bound \(\lceil\frac{n(n+2)}{4}\rceil\), where n denotes the number of inner faces of G.

Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

We define and investigate extension groups in the context of Arakelov geometry. The “arithmetic extension groups” Ext ˆ X i ( F , G ) we introduce are extensions by groups of analytic types of the usual extension groups Ext X i ( F , G ) attached to O X -modules F and G over an arithmetic scheme X. In this paper, we focus on the first arithmetic extension group Ext ˆ X 1 ( F , G ) – the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over X – and we especially consider the case when X is an “arithmetic curve”, namely the spectrum Spec O K of the ring of integers in some number field K. Then the study of arithmetic extensions over X is related to old and new problems concerning lattices and the geometry of numbers. Namely, for any two hermitian vector bundles F ¯ and G ¯ over X : = Spec O K , we attach a logarithmic size s F ¯ , G ¯ ( α ) to any element α of Ext ˆ X 1 ( F , G ) , and we give an upper bound on s F ¯ , G ¯ ( α ) in terms of slope invariants of F ¯ and G ¯ . We further illustrate this notion by relating the sizes of restrictions to points in P 1 ( Z ) of the universal extension over P Z 1 to the geometry of PSL 2 ( Z ) acting on Poincaré's upper half-plane, and by deducing some quantitative results in reduction theory from our previous upper bound on sizes. Finally, we investigate the behaviour of size by base change (i.e., under extension of the ground field K to a larger number field K ′ ): when the base field K is Q , we establish that the size, which cannot increase under base change, is actually invariant when the field K ′ is an abelian extension of K, or when F ¯ ∨ ⊗ G ¯ is a direct sum of root lattices and of lattices of Voronoi's first kind. The appendices contain results concerning extensions in categories of sheaves on ringed spaces, and lattices of Voronoi's first kind which might also be of independent interest.

Sums of Lattices and a Relational Category

We introduce a new relational category of lattices, and an analogous category of complete lattices. These categories allow us to construct sums of (complete) lattices. While previous constructions used two functors (or, for complete lattices, a single functor that had an adjoint), we need only a single functor (and no additional property when complete lattices are considered). In the finite case, the present construction is easy to visualize.

A note on semigroups, groups and geometric lattices

Let G be a closed, additive semigroup in a Hausdorff topological vector space. Then G is a group if and only if it satisfies natural convexity conditions of algebraic or geometric-topological type. This yields a characterization of the geometric lattices among the discrete, additive semigroups of Euclidean d-space \({\mathbb{E}^{d}}\) and, more generally, of direct sums of subspaces and lattices in \({\mathbb{E}^{d}}\).

On finite elements in sublattices of Banach lattices

AbstractLet E be a Banach lattice. Let H stand for a sublattice, an ideal or a band in E, and denote by Φ1(E) and Φ1(H) the ideals of finite elements in the vector lattices E and H, respectively. In this paper we first present some sufficient conditions and some counterexamples for the inclusions Φ1(H) ⊂ Φ1(E) and Φ1(E) ∩ H ⊂ Φ1(H) to hold or not. For closed ideals (and therefore bands) H there always holds Φ1(H) ⊂ Φ1(E). If H is a projection band then even PH Φ1(E) = Φ1(E) ∩ H = Φ1(H). It is proved that every finite element of E is also finite both in its Dedekind completion Ê and in its bidual space E ″. Some results concerning the finite elements in direct sums of Banach lattices are also included. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Negations and contrapositions of complete lattices

We introduce the negation CL of a complete lattice L as the concept lattice of the complementary context ( T L, M L, ≰), formed by the join-irreducible elements as objects and the meet-irreducible elements as attributes. We show that the double negation CCL is always order-embeddable in L, and that for finite lattices, the sequence ( C n L) nϵω runs into a ‘flip-flop’ (i.e., C nL ⋍ C n + 2 L for some n). Using vertical sums, we provide constructions of lattices which are isomorphic or dually isomorphic to their own negation. The only finite distributive examples among such ‘self-negative’ or ‘self-contrapositive’ lattices are vertical sums of four-element Boolean lattices. Explicitly, we determine all self-negative and all self-contrapositive lattices with less than 11 points.

Partitioning of processor arrays: a piecewise regular approach

The paper describes the systematic design of processor arrays with a given dimension and given number of processing elements. This problem is called partitioning. A solution to the partitioning problem is described for mapping a class of algorithms with piecewise regular dependence graphs, i.e. piecewise regular algorithms, onto processor arrays. These arrays are also piecewise regular, i.e. they are composed of a number of regularly connected homogenous subarrays. Partitioning deals with the division of the dependence graph of a piecewise regular algorithm into tiles and the scheduling of corresponding operations on a processor array of fixed size and dimension. Different solutions to this problem are termed partitioning schemes. Partitioning schemes may be classified into projection, multiprojection, passive and active clustering. The hereafter presented unified approach to the solution of the partitioning problem is based on the following concepts: (1) Algorithms are represented by programs. These programs can be directly interpreted as a description of hardware. (2) The concept of stepwise refinement of programs is used to solve the partitioning problem by applying a sequence of provably correct program transformations. The transformations basically involve operations on index sets. Two program transformations are introduced: (a) The EXPAND program transformation partitions the iteration space of a given program into a direct sum of lattices. The dimension of the iteration space increases. In contrary to other approaches, also nonperfect tilings may be considered. (b) Operations are scheduled on a processor array of fixed size and dimension using the REDUCE transformation. The dimension of the iteration space and thereby the dimension of the processor array is reduced. The parameters of this program transformation enable the realization of the different partitioning schemes. (3) The whole solution is embedded in the concepts of a systematic design of processor arrays.

S-pasted sum of lattices

S-pasted sum of lattices

Evaluation of Some Important Lattice Sums on NaCl and CsCl Structures

AbstractA large number of lattice sums appearing in studies on different physical properties of ionic solids are evaluated using a modified Ewald method by splitting them into sums of cubic Bravais lattices. The calculated values of the sums for the CsCl and the NaCl structure are reported in a dimensionless form. It is demonstrated that the sums of the fluorite and zincblende structures can be obtained from these sums by suitable linear combinations.