Algebra and Coalgebra on Posets

A lot of combinatorial objects have algebra and coalgebra structures and posets are important combinatorial objects. In this paper, we construct algebra and coalgebra structures on the vector space spanned by posets. Firstly, by associativity and the unitary property, we prove that the vector space with the conjunction product is a graded algebra. Then by the definition of free algebra, we prove that the algebra is free. Finally, by the coassociativity and the counitary property, we prove that the vector space with the unshuffle coproduct is a graded coalgebra.

Cyclic posets and triangulation clusters

Triangulated categories coming from cyclic posets were originally introduced by the authors in a previous paper as a generalization of the constructions of various triangulated categories with cluster structures.We give an overview, and then analyze "triangulation clusters" which are those corresponding to topological triangulations of the 2-disk. Locally finite nontriangulation clusters give topological triangulations of the "cactus space" associated to the "cactus cyclic poset".

Prinjective Ringel-Hall Algebras for Posets of Finite Prinjective Type

In the present paper we investigate prinjective Ringel-Hall algebras, for prinjective modules over incidence algebras of posers of finite prinjective type. Results we obtain are analogous to these, given by C. M. Ringel, for representations of Dynkin quivers. In particular we give a description of prinjective Ringel-Hall algebras by generators and relations.

The Folded(2D+1)-cube and Its Uniform Posets

Let F denote the folded （2D ＋ 1）-cube with vertex set X and diameter D ≥ 3. Fix x∈ X. We first define a partial order ≤ on X as follows. For y,z ∈ X let y ≤ z whenever （x,y）＋ （y,z） =- （x, z）. Let R （resp. L） denote the raising matrix （resp. lowering matrix） of P. Next we show that there exists a certain linear dependency among RL2, LRL, L2R and L for each given Q-polynomial structure of F. Finally, we determine whether the above linear dependency structure gives this poser a uniform structure or strongly uniform structure.

基于目标权重和信息增量的传感器管理方法

信息增量最大化只是传感器资源分配的必要条件而并非充分条件,为了同时满足传感器资源分配的充分条件,本文提出了一种基于目标权重和信息增量相结合的方法.该方法在利用交互式多模型(IMM)获取信息增量的基础上,结合由部分排序集合(POSET)方法而得出的目标所需的检测、跟踪和分类等任务的优先级别来对传感器资源进行分配.仿真结果表明,与仅考虑信息增量最大化的方法相比,该方法不仅行之有效而且更能合理地分配有限的传感器资源.

一些量子结构的构造和Fuzzy有效空间

给出了Fuzzy有效空间的概念.利用偏序集上的特殊模糊子集构造了effect algebras,σ-effect algebras,orthoalgebras,orthomodular posets和σ-orthomodular posets.借助Fuzzy有效空间,得到了带有强序决定态系统的lattice effect algebras的一个表现定理.

On the Maps Between Antimatroids

In this paper, the definitions of the most common and elementary mappings between matroids are extended to antimatroids first. Then the poset theory is used to find out the fiats of an antimatroid and obtain all of strong maps for a given antimatroid. Besides, the poset theory is also used to deal with the relationships among the mappings between antimatroids. All the discussion is connected with poset theory. This claims that poset theory is an important tool for the study of antimatroid theory.

Quantum Logic and Geometric Quantization

We assume that M is a phase space and H an Hilbert space yielded by a quantization scheme. In this paper we consider the set of all “experimental propositions” of M and we look for a model of quantum logic in relation to the quantization of the base manifold M. In particular we give a new interpretation about previous results of the author in order to build an “asymptotics quantum probability space” for the Hilbert lattice L(H).

Toric Heaps, Cyclic Reducibility, and Conjugacy in Coxeter Groups

In 1986, G.X. Viennot introduced the theory of heaps of pieces as a visualization of Cartier and Foata’s “partially commutative monoids”. These are essentially labeled posets satisfying a few additional properties, and one natural setting where they arise is as models of reduced words in Coxeter groups. In this paper, we introduce a cyclic version of a heap, which loosely speaking, can be thought of as taking a heap and wrapping it into a cylinder. We call this object a toric heap, because we formalize it as a labeled toric poset, which is a cyclic version of an ordinary poset. Defining the category of toric heaps leads to the notion of certain morphisms such as toric extensions. We study toric heaps in Coxeter theory, because a cyclic shift of a reduced word is simply a conjugate by an initial or terminal generator. As such, we formalize and study a framework that we call cyclic reducibility in Coxeter theory, which is closely related to conjugacy. We introduce what it means for elements to be torically reduced, which is a stronger condition than simply being cyclically reduced. Along the way, we encounter a new class of elements that we call torically fully commutative (TFC), which are those that have a unique cyclic commutativity class, and comprise a strictly bigger class than the cyclically fully commutative (CFC) elements. We prove several cyclic analogues of results on fully commutative (FC) elements due to Stembridge. We conclude with how this framework fits into recent work in Coxeter groups, and we correct a minor flaw in a few recently published theorems.

Lattice Theory for Finite Dimensional Hilbert Space with Variables in Z_d

In this work, join and meet algebraic structure which exists in non-near-linear finite geometry are discussed. Lines in non-near-linear finite geometry ?were expressed as products of lines in near-linear finite geometry ?(where?p?is a prime). An existence of lattice between any pair of near-linear finite geometry ?of ?is confirmed. For q|d, a one-to-one correspondence between the set of subgeometry ?of ?and finite geometry ?from the subsets of the set {D(d)}?of divisors of d?(where each divisor represents a finite geometry) and set of subsystems {∏(q)}?(with variables in Zq) of a finite quantum system ∏(d)?with variables in Zd?and a finite system from the subsets of the set of divisors of d?is established.

On P-sober Spaces

In this paper,we focus on p-sober spaces and prove that(1)the To space X is p-sober if and only if the Smyth power space of X is p-sober;(2)the space X has a p-sober dcpo model if and only if X is T_(1)and p-sober;(3)every non-p-sober T_(0)space does not have a p-sobrification;(4)the T_(0)space X is sober if and only if X is p-sober and PD.

连续poset的投射像的连续性

本文用反例说明了连续poset的保定向并的投射像不必连续,指出了《Continuous Lattices and Domains》一书中定理I-2.2的错误.证明了连续poset在保定向并的核映射下的像和连续收缩核还是连续的。

On a Graph Related to Subgroups Generated by a Conjugacy Class in Finite Groups

Let G be a group,and let →/ΓC(G) be a digraph whose vertices are the nontrivial conjugacy classes of G and there is an arc from a vertex C to a vertex C' if and only if C≠C' and(C)■(C').In this paper,we characterize finite groups G whose associated digraphs →/ΓC(G) are oriented trees.

A Characterization of PSL(2,q),q=5,7

In this short note we prove that the finite non-abelian simple groups PSL(2,q),where g=5,7,are determincd by their posets of classes of isomorphic subgroups.Several interesting open problems are also formulated.

Global Rank Axioms for Poset Matroids

An excellent introduction to the topic of poset matroids is due to Barnabei, Nicoletti and Pezzoli. In this paper, we investigate the rank axioms for poset matroids; thereby we can characterize poset matroids in a “global” version and a “pseudo-global” version. Some corresponding properties of combinatorial schemes are also obtained.

Strong Sperner property of the subgroup lattice of an Abelian p-group

Let n and k be arbitrary positive integers, p a prime number and L(kn)(p) the subgroup lattice of the Abelian p-group ( /pk )n. Then there is a positive integer N( n, k) such that when p 】 N( n, k), L(kn)(p) has the strong Sperner property.

A Specialization of Prinjective Ringel-Hall Algebra and the associated Lie algebra

In the present paper we describe a specialization of prinjective Ringel-Hall algebra to 1, for prinjective modules over incidence algebras of posets of finite prinjective type, by generators and relations. This gives us a generalisation of Serre relations for semisimple Lie algebras. Connections of prinjective Ringel-Hall algebras with classical Lie algebras are also discussed.

Order Topology and Bi-Scott Topology on a Poset

In this paper, some properties of order topology and bi-Scott topology on a poset are obtained. Order-convergence in posets is further studied. Especially, a sufficient and necessary condition for order-convergence to be topological is given for some kind of posers.

Explicit Expressions of Smith's Determinant on a Poset

We introduce the meet-function and join-function on a subset of a poset and use them to give some explicit expressions of Smith’s determinant.

Graph Properties and Stratified Presentations of Partially Ordered Sets

In this paper, we introduce some new definitions such as the U＊L＊ condition to describe the zero-divisor graph G = F（P） of a poser P, and give a new and quick proof to a main result in [2, 4]. By deleting a typical vertex with least degree, we provide an algorithm for finding a maximum clique of a finite graph G. We study some properties of the zero-divisor graphs of posets concerning diameters and girths. We also provide stratified presentations of posets.