可列集合套

[1]中给出模糊集的分解定理与表现定理，指出可利用集合套H＝{H（λ）|λ∈[0,1]}刻化模糊集。本文定义可列集合套H＝{H（a）|a∈Q}，Q是（0，1）的可列稠密子集。相应地给出新的分解定理与表现定理。指出可利用可列集合套刻化的模糊集。     

VC-Dimension of Sets of Permutations

VC-dimension of a set of permutations to be the maximal k such that there exist distinct that appear in A in all possible linear orders, that is, every linear order of is equivalent to the standard order of for at least one permutation . In other words, the VC-dimension of A is the maximal k such that for some the restriction of A to contains all possible linear orders. This is analogous to the VC-dimension of a set of strings. Our main result is that there exists a universal constant C such that any set of permutations with VC-dimension 2 is of size . This is analogous to Sauer's lemma for the case of VC-dimension 2. One corollary of our main result is that any acyclic set of linear orders of is of size , (a set A of linear orders on is called acyclic if no 3 elements appear in A in all 3 orders (i,j,k), (k,i,j) and (j,k,i)). The size of the largest acyclic set of linear orders has interested researchers for many years because it is the largest number of linear orders of n alternatives such that the following is always satisfied: if each one of a set of voters chooses one of these orders as his preference then the majority relation between each two alternatives is transitive. Received August 6, 1998     

Some t-Homogeneous Sets of Permutations

Perpendicular Arrays are orderedcombinatorial structures, which recently have found applicationsin cryptography. A fundamental construction uses as ingredientscombinatorial designs and uniformly t-homogeneoussets of permutations. We study the latter type of objects. Thesemay also be viewed as generalizations of t-homogeneousgroups of permutations. Several construction techniques are given.Here we concentrate on the optimal case, where the number ofpermutations attains the lower bound. We obtain several new optimalsuch sets of permutations. Each example allows the constructionof infinite families of perpendicular arrays.     

Spaces in Which Countable Closed Sets Have Countable Character

It is proved that in a T ₃ space countable closed sets have countable character if and only if the set of limit point of the space is a countable compact set and every compact set is of countable character. Also, it is shown that spaces where countable sets have countable character are WN-spaces and are very close to M-spaces. Finally, some questions of Dai and Lia are discussed and some questions are proposed.     

Partially Well-Ordered Closed Sets of Permutations

It is known that the pattern containment order on permutations is not a partial well-order. Nevertheless, many naturally defined subsets of permutations are partially well-ordered, in which case they have a strong finite basis property. Several classes are proved to be partially well-ordered under pattern containment. Conversely, a number of new antichains are exhibited that give some insight as to where the boundary between partially well-ordered and not partially well-ordered classes lies.     

Generating Multiset Permutations in Constant Time

An algorithm is presented that generates multiset permutations taking constant time between each permutation.     

Generating Functions for Permutations Avoiding a Consecutive Pattern

Given a permutation τ of length j, we say that a permutation σ has a τ-match starting at position i, if the elements in positions i, i+1, . . . , i+j−1 in σ have the same relative order as the elements of τ. We have been able to take advantage of the results of Mendes and Remmel [1] to obtain a generating function for the number of permutations with no τ-matches for several new classes of permutations. These new classes include a large class of permutations which are shuffles of an increasing sequence 1 2 · · · n with an arbitrary permutation σ of the integers {n + 1, . . . , n + m}. Finally we give a formula for the generating function for the number of permutations which have no 1 3 2 4-matches.     

Generalized Vector Variational Inequalities over Countable Product of Sets

In this paper, we consider vector variational inequalities with set-valued mappings over countable product sets in a real Banach space setting. By employing concepts of relative pseudomonotonicity, we establish several existence results for generalized vector variational inequalities and for systems of generalized vector variational inequalities. These results strengthen previous existence results which were based on the usual monotonicity type assumptions     

Generalized Vector Variational Inequalities over Countable Product of Sets

In this paper, we consider vector variational inequalities with set-valued mappings over countable product sets in a real Banach space setting. By employing concepts of relative pseudomonotonicity, we establish several existence results for generalized vector variational inequalities and for systems of generalized vector variational inequalities. These results strengthen previous existence results which were based on the usual monotonicity type assumptions     

Countable Random Sets: Uniqueness in Law and Constructiveness

The first part of this article deals with theorems on uniqueness in law for σ-finite and constructive countable random sets, which in contrast to the usual assumptions may have points of accumulation. We discuss and compare two approaches on uniqueness theorems: first, the study of generators for σ-fields used in this context and, secondly, the analysis of hitting functions. The last section of this paper deals with the notion of constructiveness. We prove a measurable selection theorem and a decomposition theorem for constructive countable random sets, and study constructive countable random sets with independent increments.     

Generating Self-Map Monoids of Infinite Sets

Let be a countably infinite set, the group of permutations of , and the monoid of self-maps of . Given two subgroups , let us write if there exists a finite subset such that the groups generated by and are equal. Bergman and Shelah showed that the subgroups which are closed in the function topology on S fall into exactly four equivalence classes with respect to . Letting denote the obvious analog of for submonoids of E, we prove an analogous result for a certain class of submonoids of E, from which the theorem for groups can be recovered. Along the way, we show that given two subgroups which are closed in the function topology on S, we have if and only if (as submonoids of E), and that for every subgroup (where denotes the closure of G in the function topology in S and its closure in the function topology in E).     

End-Faithful Spanning Trees of Countable Graphs with Prescribed Sets of Rays

We prove that a countable connected graph has an end-faithful spanning tree that contains a prescribed set of rays whenever this set is countable, and we show that this solution is, in a certain sense, the best possible. This improves a result of Hahn and Siran [2, Theorem 1]     

A New Notion of Transitivity for Groups and Sets of Permutations

Let = {1, 2, ..., n} where n 2. The shape of an ordered setpartition P = (P₁, ..., P_k) of is the integer partition =(₁, ..., _k) defined by _i = |P_i|. Let G be a group of permutationsacting on . For a fixed partition of n, we say that G is -transitiveif G has only one orbit when acting on partitions P of shape. A corresponding definition can also be given when G is justa set. For example, if = (n – t, 1, ..., 1), then a -transitivegroup is the same as a t-transitive permutation group, and if = (n – t, t), then we recover the t-homogeneous permutationgroups. We use the character theory of the symmetric group S_n to establishsome structural results regarding -transitive groups and sets.In particular, we are able to generalize a celebrated resultof Livingstone and Wagner [Math. Z. 90 (1965) 393–403]about t-homogeneous groups. We survey the relevant examplescoming from groups. While it is known that a finite group ofpermutations can be at most 5-transitive unless it containsthe alternating group, we show that it is possible to constructa nontrivial t-transitive set of permutations for each positiveinteger t. We also show how these ideas lead to a combinatorialbasis for the Bose–Mesner algebra of the association schemeof the symmetric group and a design system attached to thisassociation scheme.     

Fuzzy集的势与可数Fuzzy基数

本文首先定义了Ｆｕｚｚｙ集的等势关系和Ｆｕｚｚｙ集的势，并讨论了有关Ｆｕｚｚｙ集的势的若干命题，然后在自然数集上定义了可数Ｆｕｚｚｙ基数。最后给出了确定可数论域上Ｆｕｚｚｙ集的Ｆｕｚｚｙ基数的方法，并对其若干性质进行了讨论。上述定义均以经典集论中相应定义为特款。     

Generating Sets of Completely 0-Simple Semigroups

ABSTRACT

A formula for the rank of an arbitrary finite completely 0-simple semigroup, represented as a Rees matrix semigroup ?⁰[G; I, Λ; P], is given. The result generalizes that of Ru?kuc concerning the rank of connected finite completely 0-simple semigroups. The rank is expressed in terms of |I|, |Λ|, the number of connected components k of P, and a number r _min, which we define. We go on to show that the number r _min is expressible in terms of a family of subgroups of G, the members of which are in one-to-one correspondence with, and determined by the nonzero entries of, the components of P. A number of applications are given, including a generalization of a result of Gomes and Howie concerning the rank of an arbitrary Brandt semigroup B(G,{1,…,n}).     

The Joint Distribution of Descent and Major Index over Restricted Sets of Permutations

We compute the joint distribution of descent and major index over permutations of {1,..., n} with no descents in positions {n − i, n − i + 1, ... , n − 1} for fixed i ≥ 0. This was motivated by the problem of enumerating symmetrically constrained compositions and generalizes Carlitz’s q-Eulerian polynomial. Received December 19, 2006     

Generating and Enumerating Digitally Convex Sets of Trees

A set S of vertices in a graph G with vertex set V is digitally convex if for every vertex \(v \in V\), \(N[v] \subseteq N[S]\) implies \(v \in S\). We show that a vertex belongs to at most half of the digitally convex sets of a graph. Moreover, a vertex belongs to exactly half of the digitally convex sets if and only if it is simplicial. An algorithm that generates all digitally convex sets of a tree is described and sharp upper and lower bounds for the number of digitally convex sets of a tree are obtained. A closed formula for the number of digitally convex sets of a path is derived. It is shown how a binary cotree of a cograph can be used to enumerate its digitally convex sets in linear time.     

Virasoro李代数的子代数间的同构及生成元

证明了无中心Virasoro李代数的有限维子代数同构的充分必要条件,证明了两个元素di,dj作为生成元的充分必要条件,找出了几组互相同构的无限维真子代数,研究了他们的极大性,单性以及其它性质.     

Generating Systems of Sets and Quotients of Surjeetive Operator Ideals

The paper gives a characterization of right-hand quotients A = A ₁ ? A ^?1₂ of surjective operator ideals A ₁ and A ₂ which refers to so called “generating systems of sets”. Simultaneously the corresponding “inversion formulas” A ₁ = ( A ? A ₂)^S and A ₂ = A ^?1 ? A ₁ are investigated.