Largest induced suborders satisfying the chain condition

For a finite ordered set P, let c(P) denote the cardinality of the largest subset Q such that the induced suborder on Q satisfies the Jordan-Dedekind chain condition (JDCC), i.e., every maximal chain in Q has the same cardinality. For positive integers n, let f(n) be the minimum of c(P) over all ordered sets P of cardinality n. We prove:      

The dimension of suborders of the Boolean lattice

We consider the order dimension of suborders of the Boolean latticeB _n . In particular we show that the suborder consisting of the middle two levels ofB _n dimension at most of 6 log₃ n. More generally, we show that the suborder consisting of levelss ands+k ofB _n has dimensionO(k ² logn).The research of the second author was supported by Office of Naval Research Grant N00014-90-J-1206.The research of the third author was supported by Grant 93-011-1486 of the Russian Fundamental Research Foundation.     

The jump number of suborders of the power set order

Let P be an ordered set induced by several levels of a power set. We give a formula for the jump number of P and show that reverse lexicographic orderings of P are optimal. The proof is based on an extremal set result of Frankl and Kalai.     

拟半连续格和交半连续格

作为半连续格的推广, 引入了拟半连续格的概念. 讨论了拟半连续格的基本性质. 在拟半连续格上得到了类似于拟连续偏序集的一些主要结果. 同时探讨了半连续格、拟半连续格、交半连续格、交连续格、强连续格几种不同结构之间的关系. 最后,讨论了半连续函数空间仍是半连续格的条件.     

Isometrical Embeddings of Lattices into Geometric Lattices

A lattice is said to be finite height generated if it is complete and every element is the join of some elements of finite height. Extending former results by Grätzer and Kiss (Order 2:351–365, 1986) on finite lattices, we prove that every finite height generated algebraic lattice that has a pseudorank function is isometrically embeddable into a geometric lattice.     

Smooth格和强Smooth格

引入了强smooth格的概念,讨论了smooth格与强smooth格的一些基本性质,证明了强smooth格可用保任意交和Scott闭集之并的映射嵌入到某方体[0,1]X之中.     

半连续格和半代数格的映射性质

研究了半连续格及半代数格的一些映射性质,讨论了强连续格的函数空间,给出了强连续格到方体的嵌入定理.     

蕴涵格的同余关系及熵蕴涵格

在蕴涵格中引入了蕴涵滤子的概念 ,讨论了蕴涵滤子的一些基本性质 ,并由此建立了由素蕴涵滤子决定的同余关系及其商蕴涵格 ,以便为Fuzzy推理建立严格的逻辑基础作些必要的准备 .     

Embedding Finite Lattices into Finite Biatomic Lattices

For a class C of finite lattices, the question arises whether any lattice in C can be embedded into some atomistic, biatomic lattice in C. We provide answers to the question above for C being, respectively,– the class of all finite lattices;– the class of all finite lower bounded lattices (solved by the first author's earlier work);– the class of all finite join-semidistributive lattices (this problem was, until now, open).We solve the latter problem by finding a quasi-identity valid in all finite, atomistic, biatomic, join-semidistributive lattices but not in all finite join-semidistributive lattices.     

Lattices That are Embeddable in Suborder Lattices

Various types of lattices are embedded in suborder lattices of posets possessing certain properties. In particular, it is shown that the class of lattices isomorphic to sublattices of suborder lattices of posets of length at most n is a variety, for any n < ω.Supported by INTAS-YSF grant 2001/1-65, by RFBR-DFG grant No. 01-01-04003, by RF Ministry of Education grant No. E02-1.0-32, by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2112.2003.1, and by a grant from the Russian Science Support Foundation.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 483–511, July–August, 2005.     

Banach格上正则算子格

本文首先对 Ｂａｎａｃｈ格 Ｅ给出了条件,使得对任意非 Ｄｅｄｅｋｉｎｄ　σ－完备的Ｂａｎａｃｈ格Ｆ,正则算子空间Ｌ~ｒ（Ｅ,Ｆ）均是一Ｒｉｅｓｚ空间．其次对Ｂａｎａｃｈ格Ｆ给出了一些刻划,使之每个由Ｌ_ｐ－空间到Ｆ内的连续线性算子均是正则的．一些相关结果也得以讨论．     

真蕴涵格与优蕴涵格

研究蕴涵格的性质及其结构。借助数学软件给出8阶及以下蕴涵格的完全分类,说明了真蕴涵格与优蕴涵格的存在性,并纠正了多个文献中关于蕴涵格的一些错误结论。同时,引入局部有限蕴涵格和次结合蕴涵格的概念,证明了局部有限蕴涵格是优蕴涵格。     

剩余格与正则剩余格的特征定理

本文进一步研究了具有广泛应用的一类模糊逻辑代数系统——剩余格，并引入了正则剩余格的概念，对剩余格与正则剩余格的定义进行了讨论，给出了剩余格与正则剩余格的特征定理，其中包含剩余格与正则剩余格的等式特征，从而这两个格类都构成簇．本文还讨论了剩余格与正则剩余格公理系统的独立性，以及它们与相近代数结构的关系．     

The Best Extending Cover-preserving Geometric Lattices of Semimodular Lattices

In 2010,Gábor Czédli and E.Tamás Schmidt mentioned that the best cover-preserving embedding of a given semimodular lattice is not known yet [A cover-preserving embedding of semimodular lattices into geometric lattices.Advances in Mathematics,225,2455-2463(2010)].That is to say:What are the geometric lattices G such that a given finite semimodular lattice L has a cover-preserving embedding into G with the smallest |G|?In this paper,we propose an algorithm to calculate all the best extending cover...     

Crosscut-Simplicial Lattices

We call a finite lattice crosscut-simplicial if the crosscut complex of every nuclear interval is equal to the boundary of a simplex. Every interval of such a lattice is either contractible or homotopy equivalent to a sphere. Recently, Hersh and Mészáros introduced SB-labelings and proved that if a lattice has an SB-labeling then it is crosscut-simplicial. Some known examples of lattices with a natural SB-labeling include the join-distributive lattices, the weak order of a Coxeter group, and the Tamari lattice. Generalizing these three examples, we prove that every meet-semidistributive lattice is crosscut-simplicial, though we do not know whether all such lattices admit an SB-labeling. While not every crosscut-simplicial lattice is meet-semidistributive, we prove that these properties are equivalent for chamber posets of real hyperplane arrangements.     

Suslin Lattices

In their work on spreading models in Banach spaces, Dilworth et al. (Isr J Math 161:387–411, 2007) introduced the notion of a Suslin lower semi-lattice, a seemingly slight weakening of the notion of a Suslin tree. They posed several problems of a set theoretic nature regarding their notion. In this paper, we make a systematic study of the notion of Suslin lower semi-lattice, answering some of the questions raised by Dilworth, Odell, and Sari.     

Solitaire Lattices

 One of the classical problems concerning the peg solitaire game is the feasibility issue. Tools used to show the infeasibility of various peg games include valid inequalities, known as pagoda-functions, and the so-called rule-of-three. Here we introduce and study another necessary condition: the solitaire lattice criterion. While the lattice criterion is shown to be equivalent to the rule-of-three for the classical English 33-board and French 37-board as well as for any m×n board, the lattice criterion is stronger than the rule-of-three for games played on more complex boards. In fact, for a wide family of boards presented in this paper, the lattice criterion exponentially outperforms the rule-of-three. Received: February 22, 1999?Final version received: June 19, 2000     

Lattices of Games

We show that, for any set S of combinatorial games, the set of games all of whose immediate options belong to S forms a complete lattice. If every option of a game in S also lies in S, then this lattice is completely distributive.     

Lattices of subclasses

We investigate the structure of lattices of subclasses of different types; among those are relative sub(quasi)variety lattices as well as relative (finitary) subprevariety lattices. Among other results, we prove a reduction theorem for (finitary) subprevariety lattices which generalizes a result of V. A. Gorbunov. We also answer a question by D. E. Pal-chunov and provide some noncomputability properties of lattices of relative subclasses.