Lattice Of Finite Length Research Articles

A Characterization of a Semimodular Lattice

A geometric lattice is the lattice of closed subsets of a closure operator on a set which is zero-closure, algebraic, atomistic and which has the so-called exchange property. There are many profound results about this type of lattices, the most recent one of which, due to Czedli and Schimdt (Adv Math 225:2455–2463, 2010), says that a lattice L of finite length is semimodular if and only if L has a cover-preserving embedding into a geometric lattice G of the same length. The goal of our paper is to offer the following result: a lattice of finite length is semimodular if and only if every cell in L is a 4-element Boolean lattice and the 7-element non-distributive atomistic lattice having 3 atoms is not a cover-preserving sublattice of L.

Multi-Dimensional Random Walks and Integrable Phase Models

We consider random multi-dimensional lattice walks bounded by a hyperplane, calling them walks over multi-dimensional simplicial lattices. We demonstrate that generating functions of these walks are dynamical correlation functions of a certain type of exactly solvable quantum phase models describing strongly correlated bosons on a chain. Walks over oriented lattices are related to the phase model with a non-Hermitian Hamiltonian, while walks over disoriented ones are related to the model with a Hermitian Hamiltonian. The calculation of the generating functions is based on the algebraic Bethe Ansatz approach to the solution of integrable models. The answers are expressed through symmetric functions. Continuous-time quantum walks bounded by a onedimensional lattice of finite length are also studied. Bibliography: 40 titles.

Planar Graded Lattices and the c 1-Median Property

Let L be a lattice of finite length, ξ = (x1,…, xk)∈Lk, and y ∈ L. The remotenessr(y, ξ) of y from ξ is d(y, x1)+⋯+d(y, xk), where d stands for the minimum path length distance in the covering graph of L. Assume, in addition, that L is a graded planar lattice. We prove that whenever r(y, ξ) ≤ r(z, ξ) for all z ∈ L, then y ≤ x1∨⋯∨xk. In other words, L satisfies the so-called c1-median property.

Bloch analysis of photonic lattices that incorporate vertical cavity surface-emitting laser arrays

We analyzed one-dimensional photonic lattices that incorporate mirror-modulated vertical cavity surface-emitting laser arrays utilizing the Bloch formalism. First, infinitely long arrays are considered. The in-phase mode (with a main central lobe at the far field) and antiphase mode (with two main symmetrically-located lobes at the far-field) are examined. A comparison of the modal losses of the in-phase and the antiphase modes, resulted in the discovery of regimes in which the in-phase mode is dominant. Considering lattices of finite length, we compared the results of the Bloch model to the exact solutions. It is shown that the boundary conditions in these lattices select a specific mode from the continuous spectrum in the infinite case. Consequently, the lattice's length affects the eigenmodes and the corresponding eigenvalues in a periodic manner.

Logarithmic corrections in the one-dimensional Hubbard model with attraction

The free energy in the vicinity of a half-filled band for the one-dimensional Hubbard model with attraction is calculated as a function of density (chemical potential). The expansion of the elementary excitation energies for a lattice of finite length N is accomplished with an accuracy of the order (N2 ln N)−1. An explicit expression for the free energy and the elementary excitation spectrum as functions of external field or of volume N is necessary for studying the asymptotic behavior of correlation functions.

New eigenstates of the 1-dimensional Hubbard model

Carrying out a program proposed by C.N. Yang, we prove that all “regular” (as defined in the paper) Bethe-Ansatz states of the 1-dimensional Hubbard model on a lattice of finite length L are lowest-weight vectors of an SO(4) algebra. Thus new eigenstates can be obtained by acting with the SO(4) raising operators on the Bethe-Ansatz states. In a following publication we will show that the SO(4) structure in combination with the Bethe Ansatz leads to a complete set of eigenfunctions for the 1-dimensional Hubbard model (asymptotically for large but finite lattice lengths L).

Cover-preserving embedding

2. Preliminaries All lattices considered in this paper are finite modular lattices. First we need some definitions (see G. Grs [3]). We shall say that a/b is perspective to c/d, and write a/b.~ c/d if either b = hAd, c= aVdor a = bVc, d= bAc. In the first case we write a/b/z c/d and we write a/b ~x~ c/d in the second case. If for some natural number n there exist a/b = eo/fo ~ el/fl ..... e,/f~ = c/d, we shall say that a/b is projective to c/d. This will be denoted by a/b ~ c/d. We write c/d'~ a/b iff b < d and c = a V d; similarly, c/d/z~ a/b iff c < a and d = b A e. If c/d/z a/b or c/d "~ a/b, then c/d is weakly perspective into a/b, in symbols c/d .~ a/b. If for some natural number n and c/d = eo/fo, el/fl,..., e,/f, = a/b we have ei/fi "~ ei+l/fi+l, i = 0, ..., n - 1 then c/d is weakly projective into a/b, in symbol, c/d ~w a/b. The well-known Hall-Dilworth gluing of a filter and ideal was extended in [4] by Ch. Herrmann. This construction is called the S-glued sum, where S is a lattice of finite length. The definition is based on the following concept. Let S be a lattice.

On the contraction of vectorial lattice representations

For modular lattices of finite length, vector space representations are shown to give rise to contracted representations of homomorphic images

Interval order and semiorder lattices

When a set of closed intervals of the reals is partially ordered by decreeing that A<B when A lies strictly to the left of B, the resulting structure is called an interval order. Semiorders may be viewed as interval orders that arise from closed intervals having a fixed length. The paper initiates a careful study of interval orders and semiorders that happen also to be lattices. A structure theory is obtained for a class of interval order lattices that includes all such lattices of finite length. Characterizations are given of when these lattices are modular or distributive, as well as when they are semiorders. The theory is of some interest because the completion by cuts of an interval order is necessarily an interval order lattice. Though it is shown that the completion by cuts of a semiorder need not be a semiorder, necessary and sufficient conditions are given for a lattice of finite length to be isomorphic to the completion by cuts of a semiorder.

Meet-regular intervals in lattices of finite length

Meet-regular intervals in lattices of finite length

Geometric exchange properties in lattices of finite length

We investigate geometric exchange properties in lattices of finite length that generalize the Steinitz exchange property of finite-dimensional vector spaces. In particular, we show that a stronger version of MacLane's exchange property for semimodular lattices is equivalent to the join-symmetric exchange property of Gaskill and Rival for modular lattices and, furthermore, that this exchange property characterizes strong semimodular lattices. An analogue of the basis exchange property for matroids is considered and seen to distinguish strong lattices in the class of semimodular lattices.

Semimodularity in lattices of finite length

For a lattice L of finite length we denote by J( L) the set of all join-irreducible elements (≠0) of L. By u′ we mean the uniquely determined lower cover of an element uϵJ( L). Our main result is the following theorem: A lattice L of finite length is (upper) semimodular if and only if it satisfies the exchange property (EP): c⩽ b∨ u and c≰b∨u′ imply u⩽ b∨ c∨ u′ ( b, cϵL; uϵJ( L)).

On complements in lattices of finite length

Let L be a lattice of finite length. It is shown that if L contains no 3-element interval then L is relatively complemented. Also, if for every x ε L the set C x =y ε L ¦ y covers x satisfies V C x = 1, then L is complemented.

Sublattices of Modular Lattices of Finite Length

It is well-known that the join-irreducible elements J(L) and the meet-irreducible elements M(L) of a lattice L of finite length play a central role in its arithmetic and, especially, in the case that L is distributive. In [3] it was shown that the quotient set Q(L) = {b/a | a ∊ J(L), b ∊ M(L), a ≤ b} plays a somewhat analogous role in the study of the sublattices of L. Indeed, in a lattice L of finite length, if S is a sublattice of L then S = L — ∪b/a∊A [a, b] for some A ⊆ Q(L). Furthermore, the converse actually characterizes finite distributive lattices [3].

Erzeugendensysteme und der A-Dimensionssatz in Atomaren Modularen Verbänden

In atomic modular lattices a special dimension can be introduced by atoms. The question was, whether the dimensional formula holds. Trying to prove this leads to a problem, formulated by Olaf Tamaschke (see [5]). It is shown here that an additional property is necessary, called atomic closed (atomar abgeschlossen). With this quality some new theorems can be proved. For instance: lattices not being atomic closed contain a special sublattice, and in lattices of finite length this property is equivalent to the validity of the dimensional formula.