Poset Ramsey Numbers for Boolean Lattices

Order - For each positive integer n, let Qn denote the Boolean lattice of dimension n. For posets $P, P^{prime }$ , define the poset Ramsey number $R(P,P^{prime })$ to be the least N such that...     

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.     

Incomparability and Intersection Properties of Boolean Interval Lattices and Chain Posets

In a canonical way, we establish an AZ-identity (see [2]) and its consequences, the LYM-inequality and the Sperner property, for the Boolean interval lattice. Furthermore, the Bollobas inequality for the Boolean interval lattice turns out to be just the LYM-inequality for the Boolean lattice. We also present an Intersection Theorem for this lattice.Perhaps more surprising is that by our approach the conjecture of P. L. Erdöset al.[7] and Z. Füredi concerning an Erdös–Ko–Rado-type intersection property for the poset of Boolean chains could also be established. In fact, we give two seemingly elegant proofs.     

M－连续格到Hilbert方体的嵌入

本文主要讨论Ｍ－连续格到Ｈｉｌｂｅｒｔ方体的嵌入问题．我们建立了Ｍ－连续格的次直积表示理论，推广并统一了Ｒａｎｅｙ，Ｂｒｕｎｓ，Ｌａｗｓｏｎ，Ｂａｎｄｅｌｔ和Ｅｒｎｅ等人的相应工作．Ｒｅｎａｙ与Ｂｒｕｎｓ的经典方法是建立在对相应的弱辅助关系的极大完备链作深入分析之上的，富于技巧性，且有局限性．与之相比，本文所使用的方体则相当朴素而自然，但却能处理更为广泛的情形．     

Large Sets of Lattices without Order Embeddings

Let I and μ be an infinite index set and a cardinal, respectively, such that |I| ≤ μ and, starting from ?₀, μ can be constructed in countably many steps by passing from a cardinal λ to 2^λ at successor ordinals and forming suprema at limit ordinals. We prove that there exists a system X = {L_i: i ∈ I} of complemented lattices of cardinalities less than |I| such that if i, j ∈ I and φ: L_i → L_j is an order embedding, then i = j and φ is the identity map of L_i. If |I| is countable, then, in addition, X consists of finite lattices of length 10. Stating the main result in other words, we prove that the category of (complemented) lattices with order embeddings has a discrete full subcategory with |I| many objects. Still in other words, the class of these lattices has large antichains (that is, antichains of size |I|) with respect to the quasiorder “embeddability.” As corollaries, we trivially obtain analogous statements for partially ordered sets and semilattices.     

M－连续格到Hilbert方体的嵌入

本文主要讨论Ｍ－连续格到Ｈｉｌｂｅｒｔ方体的嵌入问题．我们建立了Ｍ－连续格的次直积表示理论，推广并统一了Ｒａｎｅｙ，Ｂｒｕｎｓ，Ｌａｗｓｏｎ，Ｂａｎｄｅｌｔ和Ｅｒｎｅ等人的相应工作．Ｒｅｎａｙ与Ｂｒｕｎｓ的经典方法是建立在对相应的弱辅助关系的极大完备链作深入分析之上的，富于技巧性，且有局限性．与之相比，本文所使用的方体则相当朴素而自然，但却能处理更为广泛的情形．     

Boolean Algebras and Distributive Lattices Treated Constructively

Some aspects of the theory of Boolean algebras and distributive lattices–in particular, the Stone Representation Theorems and the properties of filters and ideals–are analyzed in a constructive setting.     

Boolean Topological Distributive Lattices and Canonical Extensions

This paper presents a unified account of a number of dual category equivalences of relevance to the theory of canonical extensions of distributive lattices. Each of the categories involved is generated by an object having a two-element underlying set; additional structure may be algebraic (lattice or complete lattice operations) or relational (order) and, in either case, topology may or may not be included. Among the dualities considered is that due to B. Banaschewski between the categories of Boolean topological bounded distributive lattices and the category of ordered sets. By combining these dualities we obtain new insights into canonical extensions of distributive lattices. The second author was supported by Slovak grants VEGA 1/3026/06 and APVV-51-009605.     

Constructing Ramsey graphs from Boolean function representations

Explicit construction of Ramsey graphs or graphs with no large clique or independent set, has remained a challenging open problem for a long time. While Erdös’ probabilistic argument shows the existence of graphs on 2n vertices with no clique or independent set of size 2 ⁿ , the best explicit constructions achieve a far weaker bound. There is a connection between Ramsey graph constructions and polynomial representations of Boolean functions due to Grolmusz; a low degree representation for the OR function can be used to explicitly construct Ramsey graphs [17,18]. We generalize the above relation by proposing a new framework. We propose a new definition of OR representations: a pair of polynomials represent the OR function if the union of their zero sets contains all points in {0, 1} ⁿ except the origin. We give a simple construction of a Ramsey graph using such polynomials. Furthermore, we show that all the known algebraic constructions, ones to due to Frankl-Wilson [12], Grolmusz [18] and Alon [2] are captured by this framework; they can all be derived from various OR representations of degree O(√n) based on symmetric polynomials. Thus the barrier to better Ramsey constructions through such algebraic methods appears to be the construction of lower degree representations. Using new algebraic techniques, we show that better bounds cannot be obtained using symmetric polynomials.     

Hamiltonian Cycles and Symmetric Chains in Boolean Lattices

Let B(n) be the subset lattice of \({\{1,2,\dots, n\}.}\) Sperner’s theorem states that the width of B(n) is equal to the size of its biggest level. There have been several elegant proofs of this result, including an approach that shows that B(n) has a symmetric chain partition. Another famous result concerning B(n) is that its cover graph is hamiltonian. Motivated by these ideas and by the Middle Two Levels conjecture, we consider posets that have the Hamiltonian Cycle–Symmetric Chain Partition (HC-SCP) property. A poset of width w has this property if its cover graph has a hamiltonian cycle which parses into w symmetric chains. We show that the subset lattices have the HC-SCP property, and we obtain this result as a special case of a more general treatment.     

On the Dual Ramsey Property for Finite Distributive Lattices

The class of finite distributive lattices, as many other natural classes of structures, does not have the Ramsey property. It is quite common, though, that after expanding the structures with appropriately chosen linear orders the resulting class has the Ramsey property. So, one might expect that a similar result holds for the class of all finite distributive lattices. Surprisingly, Kechris and Soki? have proved in 2012 that this is not the case: no expansion of the class of finite distributive lattices by linear orders satisfies the Ramsey property. In this paper we prove that the class of finite distributive lattices does not have the dual Ramsey property either. However, we are able to derive a dual Ramsey theorem for finite distributive lattices endowed with a particular linear order. Both results are consequences of the recently observed fact that categorical equivalence preserves the Ramsey property.     

IMTL代数上的素布尔滤子格

本文研究IMTL代数M上的索布尔滤子的运算性质。令FB（M）为膨上全体素布尔滤子集，FB（M）=FB（M）∪{φ}，通过在集合FB（M）引进格并、交运算和逆序对合对应，证明了FB（M）构成一个拟布黎代数。进一步在FB（M）可定义一个伴随对，证明FB（M）也构成一个剩余格。     

The Width of Random Subsets of Boolean Lattices

Suppose we toss an independent coin with probability of success p for each subset of [n]={1,…,n}, and form the random hypergraph (n,p) by taking as hyperedges the subsets with successful coin tosses. We investigate the cardinality of the largest Sperner family contained in (n,p). We obtain a sharp result for the range of p=p(n) in which this Sperner family has cardinality comparable to the cardinality of (n,p).     

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.     

Group Embeddings with Algorithmic Properties

We show that every countable group H with solvable word problem can be subnormally embedded into a 2-generated group G which also has solvable word problem. Moreover, the membership problem for H < G is also solvable. We also give estimates of time and space complexity of the word problem in G and of the membership problem for H < G.     

Embeddings into Orthomodular Lattices with Given Centers,State Spaces and Automorphism Groups

We prove that, given a nontrivial Boolean algebra B, a compact convex set S and a group G, there is an orthomodular lattice L with the center isomorphic to B, the automorphism group isomorphic to G, and the state space affinely homeomorphic to S. Moreover, given an orthomodular lattice J admitting at least one state, L can be chosen such that J is its subalgebra.     

Ramsey Properties of Finite Posets

An important problem in topological dynamics is the calculation of the universal minimal flow of a topological group. When the universal minimal flow is one point, we say that the group is extremely amenable. For the automorphism group of Fra?ssé structures, this problem has been translated into a question about the Ramsey and ordering properties of certain classes of finite structures by Kechris et al. (Geom Funct Anal 15:106–189, 2005). Using the Schmerl list (Schmerl, Algebra Univers 9:317–321, 1979) of Fra?ssé posets, we consider classes of finite posets with arbitrary linear orderings and linear orderings that are linear extensions of the partial ordering. We provide classification of each of these classes according to their Ramsey and ordering properties. Additionally, we extend the list of extremely amenable groups as well as the list of metrizable universal minimal flows.