Invariant sublattices for positive operators

There are, by now, many results which guarantee that positive operators on Banach lattices have non-trivial closed invariant sublattices. In particular, this is true for every positive compact operator. Apart from some results of a general nature, in this paper we present several examples of positive operators on Banach lattices which do not have non-trivial closed invariant sublattices. These examples include both AM-spaces and Banach lattices with an order continuous norm and which are and are not atomic. In all these cases we can ensure that the operators do possess non-trivial closed invariant subspaces.     

Some Applications of Rademacher Sequences in Banach Lattices

We give several applications of Rademacher sequences in abstract Banach lattices. We characterise those Banach lattices with an atomic dual in terms of weak* sequential convergence. We give an alternative treatment of results of Rosenthal, generalising a classical result of Pitt, on the compactness of operators from L^p into L^q. Finally we generalise earlier work of ours by showing that, amongst Banach lattices F with an order continuous norm, those having the property that the linear span of the positive compact operators fromE into F is complete under the regular norm for all Banach lattices E are precisely the atomic lattices.     

STML(L)范畴的若干结果

本文讨论了由(?)-smooth拓扑分子格和(?)-smooth连续的广义序同态所构成的STML((?)) 范畴的若干结果,证明了范畴STML((?))是拓扑范畴,范畴TML是范畴STML((?))的双反射满子范畴,以及两个伴随定理.     

Strong semimodular lattices and Frankl's conjecture

In this paper, we show that Frankl's conjecture holds for strong semimodular lattices. The result is the first step to deal with the case of upper semimodular lattices. Received May 30, 2000; accepted in final form August 14, 2000.     

Lattices with Complemented Tolerance Lattice

We characterize lattices with a complemented tolerance lattice. As an application of our results we give a characterization of bounded weakly atomic modular lattices with a Boolean tolerance lattice.     

Cover-preserving embeddings of distributive lattices

In this paper we investigate the Cover-Preserving-Embedding-Property, for finite distributive lattices with respect to the variety of modular lattices and give a sufficient condition. Received September 27, 1995; accepted in final form April 6, 1998.     

对格上Erceg式p．q．（p．）度量的注记

本文的主要结果是大幅度地简化了Ｅｒｃｅｇ引入的不分明度量的无点化定义，建立了相应的点式刻画．     

Superrigidity for irreducible lattices and geometric splitting

We prove general superrigidity results for actions of irreducible lattices on CAT spaces, first in terms of the ideal boundary, and then for the intrinsic geometry (also for infinite-dimensional spaces). In particular, one obtains a new and self-contained proof of Margulis' superrigidity theorem for uniform irreducible lattices in non-simple groups. The proofs rely on simple geometric arguments, including a splitting theorem which can be viewed as an infinite-dimensional (and singular) generalization of the Lawson-Yau/Gromoll-Wolf theorem. Appendix A gives a very elementary proof of commensurator superrigidity; Appendix B proves that all our results also hold for certain non-uniform lattices.

     

有限Chevalley群作用下的子代数轨道生成的格

讨论了一般情形Chevalley群作用下的子代数轨道生成的格.在同类型格中,研究了不同格之间的包含关系,并对格中子代数的特性及格的几何性进行了刻画.     

对称拓扑分子格的直和

文章给出了对称拓扑分子格的直和概念,给出了拓扑分子格的直和的特征,证明了对称拓扑分子格的分离性Ti(i=-1,0,1,2)及可数性CⅠ,CⅡ是可和性质.     

Interpolation on lattices generated by cubic pencils

Principal lattices are distributions of points in the plane obtained from a triangle by drawing equidistant parallel lines to the sides and taking the intersection points as nodes. Interpolation on principal lattices leads to particularly simple formulae. These sets were generalized by Lee and Phillips considering three-pencil lattices, generated by three linear pencils. Inspired by the addition of points on cubic curves and using duality, we introduce an addition of lines as a way of constructing lattices generated by cubic pencils. They include three-pencil lattices and then principal lattices. Interpolation on lattices generated by cubic pencils has the same good properties and simple formulae as on principal lattices. Dedicated to C.A. Micchelli for his mathematical contributions and friendship on occasion of his sixtieth birthday Mathematics subject classifications (2000) 41A05, 41A63, 65D05. J.M. Carnicer: Partially supported by the Spanish Research Grant BFM2003-03510, by Gobierno de Aragón and Fondo Social Europeo.     

STML(C)范畴

本文讨论由Lsmooth拓扑分子格和(L1,L2)smooth连续的广义序同态所构成的STML(C)范畴的若干结果,利用集合套的方法给出了态射的若干性质,证明了范畴STML(C)是拓扑范畴.     

有1分配格的若干种等价定义

对有单位元1的分配格,给出若干种等价定义,并根据分配格所满足的幂等律、交换律、结合律、吸收律和分配律证明了最新定义与原始定义的等价性。     

Truncated Distributive Lattices: Conceptual Structures of Simple-Implicational Theories

Logical relationships in everyday human thought are predominantly inferences with one-element premises. This becomes apparent in the practice of Formal Concept Analysis by the frequent occurrence of truncated distributive lattices as concept lattices. This paper gives a mathematization of the underlying everyday theories of logical relationships and elaborates useful mathematical results, in particular about algorithmically drawing concept lattices which correspond to the everyday logical theories. This revised version was published online in June 2006 with corrections to the Cover Date.     

On continuous noncomplete lattices

We consider noncomplete continuous and algebraic lattices and prove that finitely generated free lattices are algebraic. We also study the Lawson topology, the second most important topology in the theory of continuous domains, on finitely presented lattices. In particular, we prove that algebraic finitely presented lattices are linked bicontinuous and the Lawson topology on these lattices coincides with the interval topology. Several examples of non-distributive and noncomplete algebraic and continuous lattices are given in the paper. Received April 5, 2000; accepted in final form December 12, 2000.     

K-Theory of Noncommutative Lattices

Noncommutative lattices have been recently used as finite topological approximations in quantum physical models. As a first step in the construction of bundles and characteristic classes over such noncommutative spaces, we shall study their K-theory. We shall do it algebraically, by studying the algebraic K-theory of the associated algebras of continuous functions which turn out to be noncommutative approximately finite dimensional (AF)C^*. We also work out several examples.     

Two extension theorems. Modular functions on complemented lattices

We prove an extension theorem for modular functions on arbitrary lattices and an extension theorem for measures on orthomodular lattices. The first is used to obtain a representation of modular vector-valued functions defined on complemented lattices by measures on Boolean algebras. With the aid of this representation theorem we transfer control measure theorems, Vitali-Hahn-Saks and Nikodým theorems and the Liapunoff theorem about the range of measures to the setting of modular functions on complemented lattices.     

Generation of lattices of points for bivariate interpolation

Principal lattices in the plane are distributions of points particularly simple to use Lagrange, Newton or Aitken–Neville interpolation formulae. Principal lattices were generalized by Lee and Phillips, introducing three-pencil lattices, that is, points which are the intersection of three lines, each one belonging to a different pencil. In this contribution, a semicubical parabola is used to construct lattices of points with similar properties. For the construction of new lattices we use cubic pencils of lines and an addition of lines on them. AMS subject classification 41A05, 65D05, 41A63Research partially supported by the Spanish Research Grant BFM2003-03510, by Gobierno de Aragón and Fondo Social Europeo.     

A Note of Regularity on Completely Distributive Lattices

ＡＮｏｔｅｏｆＲｅｇｕｌａｒｉｔｙｏｎＣｏｍｐｌｅｔｅｌｙＤｉｓｔｒｉｂｕｔｉｖｅＬａｔｔｉｃｅｓ￥ＣｈｅｎＹｉｘｉａｎｇ（ＸｕｚｈｏｕＴｅａｃｈｅｒｓＣｏｏｅｇｅ，２２１００９）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｎｏｔｅ，ｔｈｅａｕｔｈｏｒｇｉｖｅｓｓ...     

Characterizations of classical orthogonal polynomials on quadratic lattices

This paper is devoted to characterizations of classical orthogonal polynomials on quadratic lattices by using a matrix approach. In this form we recover the Hahn, Geronimus, Tricomi and Bochner type characterizations of classical orthogonal polynomials on quadratic lattices. Moreover a new matrix characterization of classical ortho-gonal polynomials in quadratic lattices is presented. From the Bochner type characterization we derive the three-term recurrence relation coefficients for these polynomials.