半连续格的刻画和映射

本文讨论了半连续格的一些性质,在半连续格中引入半Scott开集族,用半Scott开集族来刻画半连续格,同时定义了半连续格之间的半连续映射,得到闭包算子的像仍是半连续格的条件.最后,研究了半连续格上的半连续映射的全体不动点之集的性质。     

Semiprime Ideals and Separation Theorems for Posets

The concept of a semiprime ideal in a poset is introduced. The relations between the semiprime (prime) ideals of a poset and the ideals of the set of all ideals of the poset are established. A result analogous to Separation Theorem is obtained in respect of semiprime ideals. Further, a generalization of Stone’s Separation Theorem for posets is obtained in respect of prime ideals. Some counterexamples are also given.      

Quasiregular and Baer lattices — II

Let L be a compactly generated multiplicative lattice with 1 compact in which every finite product of compact elements is compact and u ∈ L be a radical element. Under this hypothesis, in this paper we extend the concepts such as Baer lattices, quasiregular lattices etc. to what we respectively call Baer lattices w.r.t. u, quasiregular lattices w.r.t. u etc. and using these concepts we extend here the results proved by D. D. Anderson, C. Jayaram and P. A. Phiri. In the course of the development, we add many fruitful concepts and results. Also we study nature and characterizations of different elements.      

Order affine completeness of lattices with Boolean congruence lattices

This paper grew out from attempts to determine which modular lattices of finite height are locally order affine complete. A surprising discovery was that one can go quite far without assuming the modularity itself. The only thing which matters is that the congruence lattice is finite Boolean. The local order affine completeness problem of such lattices L easily reduces to the case when L is a subdirect product of two simple lattices L ₁ and L ₂. Our main result claims that such a lattice is locally order affine complete iff L ₁ and L ₂ are tolerance trivial and one of the following three cases occurs:

Inverse Limit of Spaces of Upper Semicontinuous Functions

ＳｕｐｐｏｓｅｔｈａｔＸｉｓａｃｏｍｐａｃｔＨａｕｓｄｏｒｆｆｓｐａｃｅ,ａｎｄＵ(Ｘ)ｄｅｎｏｔｅｓｔｈｅｓｅｔｏｆａｌｌｔｈｅｕｐｐｅｒｓｅｍｉｃｏｎｔｉｎｕｏｕｓｆｕｎｃｔｉｏｎｓｆｒｏｍＸｔｏ [0 ,1 ] .ＴｈｅｎＵ(Ｘ)ｉｓａｃｏｍｐｌｅｔｅｌａｔｔｉｃｅｕｎｄｅｒｔｈｅｐｏｉｎｔｗｉｓｅｏｒｄｅｒｉｎｇ ,ａｎｄｉｔｓｏｐｐｏｓｉｔｅｌａｔｔｉｃｅＵ(Ｘ) ｏｐｉｓａｃｏｎｔｉｎｕｏｕｓｌａｔｔｉｃｅ .ＴｈｕｓＵ(Ｘ)ｅｎｄｏｗｅｄｗｉｔｈｔｈｅｌｉｍ ｉｎｆｔｏｐｏｌｏｇｙｏｎＵ(Ｘ) ｏｐ,ｗｈｉｃ…     

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.     

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.     

Lower Semicontinuous Regularization for Vector-Valued Mappings

The paper is devoted to studying the lower semicontinuity of vector-valued mappings. The main object under consideration is the lower limit. We first introduce a new definition of an adequate concept of lower and upper level sets and establish some of their topological and geometrical properties. A characterization of semicontinuity for vector-valued mappings is thereafter presented. Then, we define a concept of vector lower limit, proving its lower semicontinuity, and furnishing in this way a concept of lower semicontinuous regularization for mappings taking their values in a complete lattice. The results obtained in the present work subsume the standard ones when the target space is finite dimensional. In particular, we recapture the scalar case with a new flexible proof. In addition, extensions of usual operations of lower and upper limits for vector-valued mappings are explored. The main result is finally applied to obtain a continuous D.C. decomposition of continuous D.C. mappings. Dedicated to Alex Rubinov in honor of his 65th birthday     

On Fixed Point and Almost Fixed Point Problems of Lower Semicontinuous Type Multivalued Mappings

In the present paper, some new almost fixed point theorems and fixed point theorems for lower semicontinuous type multivalued mappings are obtained in metrizable H-spaces.     

On the Cone of Bounded Lower Semicontinuous Functions

We prove that the cone of bounded lower semicontinuous functions defined on a Tychonoff space X is algebraically and structurally isomorphic and isometric to a convex cone contained in the cone of all bounded lower semicontinuous functions defined on the Stone-Cech compactification βX if and only if the space X is normal. We apply this theorem to the study of relationship between a class of multivalued maps and sublinear operators. Using these results, we obtain new proofs of theorems about continuous selections.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 886–902.Original Russian Text Copyright ©2005 by Yu. E. Linke.