Function Spaces of Posets with Projections

This paper investigates function spaces of structures consisting of a partially ordered set together with some directed family of projections.More precisely, given a fixed directed index set (I,), we consider triples (D,,(p _i ) _iI ) with (D,) a poset and (p _i ) _iI a monotone net of projections of D. We call them (I,)-pop's (posets with projections). Our main purpose is to study structure preserving maps between (I,)-pop's. Such homomorphisms respect both order and projections.Any (I,)-pop is known to induce a uniformity and thus a topology. The set of all homomorphisms between two (I,)-pop's turns out to form an (I,)-pop itself. We show that its uniformity is the uniformity of uniform convergence. This enables us to prove that properties such as completeness and compactness transfer to function pop's.Concerning categorical properties of (I,)-pop's, we will see that we are in a lucky situation from a computer scientist's point of view: we obtain Cartesian closed categories. Moreover, by a D -construction we get (I,)-pop's that are isomorphic to their own exponent. This yields new models for the untyped -calculus.     

An Asymptotic Ratio in the Complete Binary Tree

Let T _n be the complete binary tree of height n considered as the Hasse-diagram of a poset with its root 1 _n as the maximum element. For a rooted tree T, define two functions counting the embeddings of T into T _n as follows A(n;T)=|{S T _n  : 1 _n ∈S, S≅T}|, and B(n;T)=|{S T _n :1 _n ∉S, S≅T}|. In this paper we investigate the asymptotic behavior of the ratio A(n;T)/B(n;T), and we show that lim  _n→∞[A(n;T)/B(n;T)]=2^ℓ;−1−1, for any tree T with ℓ leaves. This revised version was published online in June 2006 with corrections to the Cover Date.     

Orbits of rational n-sets of projective spaces under the action of the linear group

Let k=Fq be a finite field. We enumerate k-rational n-sets of (unordered) points in a projective space PN over k, and we compute the generating function for the numbers of PGLN+1(k)-orbits of these n-sets. For N=1,2 we obtain a formula for these numbers of orbits as a polynomial in q with integer coefficients.     

套链分解

Let X1,X2,...,Xk be k disjoint subsets of S with the same cardinality m.Define H(m,k) = {X (C) S: X (C) Xi for 1 ≤I ≤k} and P(m,k) = {X (C) S : X ∩ Xi ≠φ for at least two Xi's}.Suppose S = Uki=1 Xi,and let Q(m,k,2) be the collection of all subsets K of S satisfying|K ∩ Xi|≥ 2 for some 1 ≤ I ≤ k.For any two disjoint subsets Y1 and Y2 of S,we define F1,j = {X (C) S : either |X ∩ Y1|≥ 1 or |X ∩ Y2|≥ j}.It is obvious that the four posers are graded posets ordered by inclusion.In this paper we will prove that the four posets are nested chain orders.     

q-Weighted log-concavity and q-product theorem on the normality of posets

We show that q-weighted log-concavity and the strict normalized matching property are preserved under the q-direct product over weighted posets. As consequences, two classes of weighted posets including the finite linear lattices are strictly q-weighted log-concave and strictly normal.     

测度拓扑和连续偏序集的刻画

研究偏序集上的测度拓扑以及与其它内蕴拓扑间的关系,利用测度拓扑刻画了偏序集的连续性.构造了反例说明存在完全分配格,其上的测度拓扑不是连续格从而不是局部紧拓扑.     

Loopless Generation of Linear Extensions of a Poset

Canfield and Williamson gave the first loopless algorithm for generating all linear extensions of a poset. It elegantly generates all signed extensions, resulting in each extension appearing somewhere with each sign, but retains only every other one independent of its sign. It uses an array for the extension. In this paper we give another loopless algorithm for generating all the linear extensions. It generates each extension only once and uses a list for the extensions.     

Weak Factorization Systems and Topological Functors

Weak factorization systems, important in homotopy theory, are related to injective objects in comma-categories. Our main result is that full functors and topological functors form a weak factorization system in the category of small categories, and that this is not cofibrantly generated. We also present a weak factorization system on the category of posets which is not cofibrantly generated. No such weak factorization systems were known until recently. This answers an open problem posed by M. Hovey.     

Dominant regions in noncrystallographic hyperplane arrangements

For a crystallographic root system, dominant regions in the Catalan hyperplane arrangement are in bijection with antichains in a partial order on the positive roots. For a noncrystallographic root system, the analogous arrangement and regions have importance in the representation theory of an associated graded Hecke algebra. Since there is also an analogous root order, it is natural to hope that a similar bijection can be used to understand these regions. We show that such a bijection does hold for type H₃ and for type I₂(m), including arbitrary ratio of root lengths when m is even, but does not hold for type H₄. We give a criterion that explains this failure and a list of the 16 antichains in the H₄ root order which correspond to empty regions.