Z-连续偏序集的表示定理

在偏序集中引入嵌入Z-基并根据嵌入Z-基建立Z-连续偏序集的表示定理.同时,我们将讨论抽象Z-基的Z-理想完备是Z-代数偏序集的条件.最后,我们深入探讨嵌入Z-基、Z-连续扩张和σz-集之间的关系.     

Z-连通连续偏序集的特征和浓度

引入了Z-连通连续偏序集的局部基和稠密子集的概念,给出了局部基的一些刻画,在此基础上定义了Z-连通连续偏序集的特征和浓度。证明了Z-连通连续偏序集的特征和浓度与Z-连通连续偏序集带上Scott拓扑时的拓扑空间的特征和浓度相等,它们分别小于Z-连通连续偏序集带上Lawson拓扑时拓扑空间的特征、浓度。     

C_Z-偏序集

我们将强Z-连续偏序集推广到了CZ-偏序集,并讨论了CZ-偏序集和强Z-连续偏序集之间的关系。同时我们定义了CZ-偏序集上的CZ-连续映射,得到CZ-偏序集在该映射下的像集仍是CZ-偏序集。最后,我们讨论了CZ-偏序集上的基及其相关性质。     

Z-连通代数偏序集及其范畴

对于Z-连通集系统,本文引入了Z-连通代数偏序集的概念,证明了Z-连通代数偏序集范畴对偶等价于强代数格范畴的一个满子范畴.     

Z-连通集系统及其范畴特征

本文引入了Z-连通集系统的概念,讨论了Z-连通连续偏序集的一系列性 质,证明了Z-连通连续偏序集范畴对偶等价于完全分配格范畴的一个满子范畴.     

Rudin性质与拟Z-连续Domain

对一般子集系统 Z,引入了 Rudin性质,给出了它的映射式刻划,作为拟连续偏序集和Z-连续偏序集的公共推广,引入了拟Z-连续Domain的概念,讨论了拟Z-连续Domain的基本性质,特别地,给出了 Rudin性质及其映射式刻划在拟 Z-连续Domain方面的若干应用,将关于拟连续偏序集的主要结果推广至了拟 Z-连续 Domain情形。     

Rudin性质与拟Z-连续Domain

对一般子集系统Z,引入了Rudin性质,给出了它的映射式刻划.作为拟连续偏序集和Z-连续偏序集的公共推广,引入了拟Z-连续Domain的概念,讨论了拟Z-连续Domain的基本性质,特别地,给出了Rudin性质及其映射式刻划在拟Z-连续Domain方面的若干应用,将关于拟连续偏序集的主要结果推广至了拟Z-连续Domain情形.     

偏序集的φ-(交)连续性及其主理想刻画

证明φ-完备偏序集是（强）P连续的当且仅当该偏序集的任一主理想是（强）φ-连续的。在φ-完备偏序集中利用φ-S集族生成f-Scott拓扑,并由此引入φ-交连续偏序集概念。证明φ-完备偏序集是P交连续的当且仅当该偏序集的任一主理想是φ-交连续的。     

偏序集的代数完备

引进一个偏序集的代数完备, 并且构造任意偏序集的一个代数完备.有最小元的并半格的代数完备正好是它的理想完备. 一个偏序集的代数完备同构于它的一个由下集作为元的完备格,并且这个完备格包含所有主理想. 基于代数完备的Galois联络的下扩张仍然是一个Galois联络.     

偏序集上一致连续性的等价刻画与性质

将一致小于关系移植到一般偏序集上,同时引入了上界小于关系,定义了偏序集的一致连续性和上界连续性.给出了一致连续偏序集的等价刻画,探讨了一致连续偏序集所具有的性质.主要结果有:(1)证明了偏序集上的一致连续性,上界连续性与s-超连续性均等价;(2)在交半格条件下,偏序集的一致连续性等价于它的每一主理想一致连续;(3)在并半格条件下,偏序集的一致连续性蕴含连续性,反之不成立;(4)一致完备的一致连续偏序集均是连续bc-dcpo,且每个主理想均为完全分配格;(5)在一致完备的条件下,一致连续性对主滤子,对闭区间,对Scott S-集以及对一致连续投射像均是可遗传的.文中也构造了若干实用的反例.     

Cartesian Closed Categories of F Z-domains

A subset systemZassigns to each partially ordered setPa certain collectionZ(P) of subsets. In this paper, a new kind of subset systems called directable subset systems is introduced. For a directable subset system Z, the concepts of F Z-way-below relation and F Z-domain are introduced. The well-known Scott topology is naturally generalized to the Z-level and the resulting topology is calledF Z-Scott topology, and the continuous functions with respect to this topology are characterized by preserving the suprema of directed Z-sets. Then, we mainly consider a generalization of the cartesian closedness of the categories DCPO of directed complete posets, BF of bifinite domains and FS ofF Sdomains to the Z-level. Corresponding to them, it is proved that, for a suitable subset system Z, the categories FZCPO ofZ-complete posets,FSFZ of finitely separated FZ-domains andBFFZ of bifiniteF Z-domains are all cartesian closed. Some examples of these categories are given.     

Noncommutative Enumeration in Graded Posets

We define a noncommutative algebra of flag-enumeration functionals on graded posets and show it to be isomorphic to the free associative algebra on countably many generators. Restricted to Eulerian posets, this ring has a particularly appealing presentation with kernel generated by Euler relations. A consequence is that even on Eulerian posets, the algebra is free, with generators corresponding to odd jumps in flags. In this context, the coefficients of the cd-index provide a graded basis.     

Filter-linkedness and its effect on preservation of cardinal characteristics

We introduce the property “F-linked” of subsets of posets for a given free filter F on the natural numbers, and define the properties “μ-F-linked” and “θ-F-Knaster” for posets in a natural way. We show that θ-F-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families.Concerning iterations of such posets, we develop a general technique to construct θ-Fr-Knaster posets (where Fr is the Frechet ideal) via matrix iterations of <θ-ultrafilter-linked posets (restricted to some level of the matrix). This is applied to prove consistency results about Cichoń's diagram (without using large cardinals) and to prove the consistency of the fact that, for each Yorioka ideal, the four cardinal invariants associated with it are pairwise different.At the end, we show that three strongly compact cardinals are enough to force that Cichoń's diagram can be separated into 10 different values.     

A Solution to the Word Problem for Free Pseudosemilattices

In this paper we solve the word problem for freepseudosemilattices by making use of particular posets whose elements are labeled by words from the absolutely free binary algebras.     

The Category of S-Posets

In this paper, we consider some category-theoretic properties of the category Pos-S of all S-posets (posets equipped with a compatible right action of a pomonoid S), with monotone action-preserving maps between them. We first discuss some general category-theoretic ingredients of Pos-S; specifically, we characterize several kinds of epimorphisms and monomorphisms. Then, we present some adjoint relations of Pos-S with Pos, Set, and Act-S. In particular, we discuss free and cofree objects. We also examine other category-theoretic properties, such as cartesian closedness and monadicity. Finally, we consider projectivity in Pos-S with respect to regular epimorphisms and show that it is the same asprojectivity, although projectives are not generally retracts of free objects over posets.     

On forbidden configurations for strong posets

The concept of strong elements in posets is introduced. Several properties of strong elements in different types of posets are studied. Strong posets are characterized in terms of forbidden structures. It is shown that many of the classical results of lattice theory can be extended to posets. In particular, we give several characterizations of strongness for upper semimodular (USM) posets of finite length. We characterize modular pairs in USM posets of finite length and we investigate the interrelationships between consistence, strongness, and the property of being balanced in USM posets of finite length. In contrast to the situation in upper semimodular lattices, we show that these three concepts do not coincide in USM posets.     

N-free posets as generalizations of series-parallel posets

N-Free posets have recently taken some importance and motivated many studies. This class of posets introduced by Grillet [8] and Heuchenne [11] are very related to another important class of posets, namely the series-parallel posets, introduced by Lawler [12] and studied by Valdes et al. [21]. This paper shows how N-free posets can be considered as generalizations of series-parallel posets, by giving a recursive construction of N-free posets. Furthermore we propose a linear time algorithm to recognize and decompose any N-free poset. This yields some very naturel problems, namely: which are the properties(such as linear time algorithm for some invariant) of series-parallel posets that are kept for N-free posets?     

The category of supercontinuous posets

In this paper, we study the order structure—supercontinuous poset, a generalization of completely distributive lattice. The Cartesian product of supercontinuous posets and some other properties of supercontinuous posets are investigated. Also, the case of superalgebraic posets are studied and some remarks on the category of supercontinuous posets are given.     

Face vectors of simplicial cell decompositions of manifolds

In this paper, we study face vectors of simplicial posets that are the face posets of cell decompositions of topological manifolds without boundary. We characterize all possible face vectors of simplicial posets whose geometric realizations are homeomorphic to the product of spheres. As a corollary, we obtain the characterization of face vectors of simplicial posets whose geometric realizations are odd-dimensional manifolds without boundary.