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1.
In this paper we study the 2-dimension of a finite poset from the topological point of view. We use homotopy theory of finite topological spaces and the concept of a beat point to improve the classical results on 2-dimension, giving a more complete answer to the problem of all possible 2-dimensions of an n-point poset.   相似文献   

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Given a finite set M of size n and a subgroup G of Sym(M), G is pertinent iff it is the automorphism group of some groupoid ??M; *??. We examine when subgroups of Sym(M) are and are not pertinent. For instance, A n , the alternating group on M, is not pertinent for n > 4. We close by indicating a natural extension of our ideas, which relates to a question of M. Gould.  相似文献   

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The poset retraction problem for a poset P is whether a given poset Q containing P as a subposet admits a retraction onto P, that is, whether there is a homomorphism from Q onto P which fixes every element of P. We study this problem for finite series-parallel posets P. We present equivalent combinatorial, algebraic, and topological charaterisations of posets for which the problem is tractable, and, for such a poset P, we describe posets admitting a retraction onto P.  相似文献   

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Given a collection S of sets, a set SS is said to be strongly maximal in S if |T?S|≤|S?T| for every TS. In Aharoni (1991) [3] it was shown that a poset with no infinite chain must contain a strongly maximal antichain. In this paper we show that for countable posets it suffices to demand that the poset does not contain a copy of posets of two types: a binary tree (going up or down) or a “pyramid”. The latter is a poset consisting of disjoint antichains Ai,i=1,2,…, such that |Ai|=i and x<y whenever xAi,yAj and j<i (a “downward” pyramid), or x<y whenever xAi,yAj and i<j (an “upward” pyramid).  相似文献   

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We show the power of posets in computational geometry by solving several problems posed on a set S of n points in the plane: (1) find the nk − 1 rectilinear farthest neighbors (or, equivalently, k nearest neighbors) to every point of S (extendable to higher dimensions), (2) enumerate the k largest (smallest) rectilinear distances in decreasing (increasing) order among the points of S, (3) given a distance δ > 0, report all the pairs of points that belong to S and are of rectilinear distance δ or more (less), covering kn/2 points of S by rectilinear (4) and circular (5) concentric rings, and (6) given a number kn/2 decide whether a query rectangle contains k points or less.  相似文献   

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Let P be a finite poset and G a group of automorphisms of P. The action of G on P can be used to define various linear representations of G, and we investigate how these representations are related to one another and to the structure of P. Several examples are analyzed in detail, viz., the symmetric group Gn acting on a boolean algebra, GLn(q) acting on subspaces of an n-dimensional vector space over GF(q), the hyperoctahedral group Bn acting on the lattice of faces of a cross-polytope, and Gn acting on the lattice Πn of partitions of an n-set. Several results of a general nature are also proved. These include a duality theorem related to Alexander duality, a special property of geometric lattices, the behavior of barycentric subdivision, and a method for showing that certain sequences are unimodal. In particular, we give what seems to be the simplest proof to date that the q-binomial coefficient k+lk has unimodal coefficients.  相似文献   

9.
灰拓扑群   总被引:1,自引:1,他引:0  
给出了灰拓扑群的定义及有关定理,在此基础上讨论灰拓扑群与Fuzzy拓扑群、一般拓各之关系,并且研究了灰拓扑群的子群。  相似文献   

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For a posetP, let Aut (P) denote the automorphism group ofP and let Fp (P) be the subposet of all fixed points of Aut (P). It is shown that for every posetP and every nontrivial groupG the posetsP satisfying Aut (P)G and Fp(P)=P form a proper class.Similarly, for a latticeL, let Aut (L) denote the automorphism group and Fp(L) the sublattice of fixed points. It is shown that ifL has more than one element andG is a nontrivial group then the latticesL for which Aut (L)G and Fp(L)=L also form a proper class. Moreover, if card (L)1 then this is still the case providingG is an infinite group. Since card (L)2 when Aut (L) is finite, this is the best possible result.With 3 FiguresThe support of the National Research Council of Canada is gratefully acknowledged.  相似文献   

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Suppose that an almost simple group G acts line transitively on a finite linear space S. Let Gx be a point stabilizer in G and suppose that G has socle T, a simple group of Lie type. In this paper we show that if TGx is a parabolic subgroup of T, then G is flag transitive on S.  相似文献   

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We give a short proof that if G is a finite group of derived length k and if G admits a fixed-point-free action of the elementary group of order 2 n , then G has a normal series of length n all of whose quotients are nilpotent of class bounded in terms of k and n only.  相似文献   

14.
Finite topological spaces, that is spaces with a finite number of points, have a wide range of applications in many areas such as computer graphics and image analysis. In this paper we study the covering dimension of a finite topological space. In particular, we give an algorithm for computing the covering dimension of a finite topological space using matrix algebra.  相似文献   

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We construct examples of localizations in the category of groups which take the Mathieu group M11 to groups of arbitrarily large cardinality which are “abelian up to finitely many generators.” The paper is part of a broader study on the group theoretic properties which are or are not preserved by localizations.  相似文献   

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We construct a full class of nilpotent groups of class 2 of an arbitrary infinite cardinality . Their centers, commutator subgroups and factors modulo the center will be the same and a homogeneous direct sum of a group of rank 1 or 2. Their automorphism groups will coincide and the factor group modulo the stabilizer could be an arbitrary group of size $\leqq$ .  相似文献   

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