偏序半群的偏序扩张

引入了偏序半群(S,·,≤)上的半拟序σ及模σ半拟链的概念.通过模σ半拟链,将S的偏序≤扩张为≤*,讨论了(S,*,≤*)是偏序半群的充分条件,并获得了若干理想的结果.特别地,得到了SPO(S)到PO(S)的两个半格同态定理.最后,还给出了S的满足某些给定条件的有限子集在≤*下成链的充要条件.     

可换BCH-代数

引入了可换BCH-代数的概念,给出了可换BCH-代数的两个充要条件.对偏序可换BCH-代数进行了讨论,给出了偏序BCH-代数是可换的两个充要条件.证明了偏序可换BCH-代数的每个分支是一个下半格,局部有界偏序可换BCH-代数的每个分支是一个格.     

S-定向完备偏序集范畴

给出定向完备偏序半群的定义,研究定向完备偏序半群在定向完备偏序集上的作用.探讨S-定向完备偏序集范畴的一些基本性质,并且证明以S-定向完备偏序集为对象,以S-Scott连续映射为态射的范畴是笛卡尔闭范畴.     

半群上的拓扑、偏序和相关Domain

在半群G上引入了半群拓扑O[G]和半群偏序≤G,研究了它们的性质和相互联系,得到如下主要结论:(1)拓扑空间(G,O[G])中开集均为偏序集(G,≤G)的下集;(2)拓扑空间(G,O[G])为T,的当且仅当O[G]是离散的,当且仅当G中任意元是幂等元;(3)在集合包含序下O[G]为代数的完全分配格;(4)若(G,O[G])是T0空间,则O[G]是偏序集(G,≤G)上的对偶Alexandrov拓扑;(5)半群G是伪有限的当且仅当偏序集(G,≤Gop)是代数Domain.     

保持序和等价关系的自然偏序变换半群

设X为一个集合,■_X为X上的全变换半群.设E是X上的一个等价关系,定义T_E(X)={f∈■_X:■(x,y)∈E,(f(x),f(y))∈E},则T_E(X)是由等价关系E所确定的■_X的子半群.本文中,所考虑的集合X是一个有限全序集,同时E是非平凡的且所有的E-类都是凸集.显然■_E(X)={f∈T_E(X):■_x,y∈X,x≤y蕴涵f(x)≤f(y)}是T_E(X)的一个子半群.我们赋予■_E(X)自然偏序并讨论何时■_E(X)中的两个元素是关于这个偏序是相关的,然后确定■_E(X)中那些关于≤是相容的元素.此外,还描述了极大(极小)元和覆盖元.     

两类剩余幺半群

主要讨论了两类可换剩余幺半群的结构及其定定理，给出并证明了一些有关的等价条件和充分必要条件，然后证明了剩余运算满足消去律的充要条件。     

偏序下的区间松弛法

§1. 引言 本文给出了求解非线性方程组 f(x)=0,f:D?R~n→R~m (1.1)在偏序下的区间松弛法,它是在[1]的基础上将区间迭代与Newton-SOR 迭代结合得到的一种便于计算且收敛较快的序区间N-SOR松弛法,也是单调N-SOR迭代法的推广.§2给出了偏序下的区间Krawczyk算子,它是区间 Newton算子的推广,同样具     

偏序可换BCH-代数分支的性质

在局部有界偏序可换BCH-代数的分支中,给出了关于一元运算N的一些性质,给出了二元运算*,∧,∨之间的一些关系式,证明了局部有界偏序可换BCH-代数的每个分支是一个分配格.     

交C-连续偏序集

利用偏序集上的半拓扑结构,引入了交C-连续偏序集概念,探讨了交C-连续偏序集的性质、刻画及与C-连续偏序集、拟C-连续偏序集等之间的关系.主要结果有:(1)交C-连续的格一定是分配格;(2)有界完备偏序集(简记为bc-poset)L是交C-连续的当且仅当对任意x∈L及非空Scott闭集S,当∨S存在时有x∧∨S=∨{x∧s:s∈S};(3)完备格是完备Heyting代数当且仅当它是交连续且交C-连续的;(4)有界完备偏序集是C-连续的当且仅当它是交C-连续且拟C-连续的;(5)获得了反例说明分配的完备格可以不是交C-连续格,交C-连续格也可以不是交连续格.     

偏序集的连通性和连通分支

本文引入了偏序集连通性的概念,给出了连通分支的构造,证明了偏序集可唯一地分解为连通分支的不交并,偏序集的连通子集是连通分支当且仅当它是分支并。     

On congruences and ideals of partially ordered quasigroups

Some results concerning congruence relations on partially ordered quasigroups (especially, Riesz quasigroups) and ideals of partially ordered loops are presented. These results generalize the assertions which were proved by Fuchs in [5] for partially ordered groups and Riesz groups.     

Splittability for Partially Ordered Sets

A topological space X is said to be splittable over a class P of spaces if for every AX there exists continuous f:XYP such that f(A)f(XA) is empty. A class P of topological spaces is said to be a splittability class if the spaces splittable over P are precisely the members of P. We extend the notion of splittability to partially ordered sets and consider splittability over some elementary posets. We identify precisely which subsets of a poset can be split along over an n-point chain. Using these results it is shown that the union of two splittability classes need not be a splittability class and a necessary condition for P to be a splittability class is given.     

The Hahn–Banach Theorem for Partially Ordered Totally Convex, Positively Convex and Superconvex Modules

The Hahn–Banach Theorem for partially ordered totally convex modules [3] and a necessary and sufficient condition for the existence of an extension of a morphism from a submodule C ₀ of a partially ordered totally convex module C (with the ordered unit ball of the reals as codomain) to C, are proved. Moreover, the categories of partially ordered positively convex and superconvex modules are introduced and for both categories the Hahn–Banach Theorem is proved.     

Building fences around the chromatic coefficients

Associated to each graph G is its chromatic polynomial f(G, t) and we associate to f(G, t) the sequence α (G) of the norms of its coefficients. A stringent partial ordering is established for such sequences. The main result is that for any graph G with q edges we have α (R_q) ≤ α (G) ≤ α (S_q), where R_q and S_q are specified graphs with q edges. This translates into a clearer view of allowable values and patterns in the chromatic coefficients. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 123–128, 1997     

On Relationships Between Semi- antichains andUnichain Coverings in Discrete Mathematics

§1.　IntroductionTherelationsbetweenchainsandantichainsinapartiallyorderedsetaretheextensionoftheconceptssuchas“selectingrepresentation”,“selectingseedings”andetc.inrealworldinpractice.Theyaretherelationstobeconsideredindesigningsystems.LetPandGbetwopartiallyorderedsets.Thepartiallyorderedrelation[1]onthemarere-spectivellyrepresentedby“≤”(Theirmeaningsmaybedifferentinpracticeindifferentpar-tiallyorderedset).ThedirectproductP×GofPandGisdefinedasfollows:P×G={(p,g)|p∈P,g∈G}.Thep…     

Axiomatizability and completeness of some classes of partially ordered polygons

We study partially ordered monoids over which a class of free (over sets and over posets), projective, and (strongly, weakly) flat partially ordered polygons is axiomatizable, complete, or model complete. Similar issues for polygons were dealt with in papers by V. Gould and A. Stepanova. Supported by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-2810.2008.1). Translated from Algebra i Logika, Vol. 48, No. 1, pp. 90–121, January–February, 2009.     

Fuzzy Ordered Filters in Implicative Semigroups

Animplicativesemilatticeisanalgebraicsystemhavingasmodelslogicalsystemsequippedwithimplicationandconjunction,butnotpossessingadisjunction.ImplicativesemilatticeswerestudiedbyW.C.Nemitz[5].In[2],T.S.BlythgeneralizedsomeresultsofW.C.Nemitz[5]byintroducingthenotionofaBrouweriansemigroup.FollowingtheideasofNemitzandBlyth,M.W.ChanandK.P.Shum[3]introducedthenotionofnegativelypartiallyorderedimplicativesemigroupsandgeneralizedsomeresultofNemitzonimplicativesemilatticestoim-plicativesemigroups…     

PARTIALLY ORDERED CONNECTIVES

We show that a coherent theory of partially ordered connectives can be developed along the same line as partially ordered quantification. We estimate the expressive power of various partially ordered connectives and use methods like Ehrenfeucht games and infinitary logic to get various undefinability results.     

Linear Discrepancy and Bandwidth

The linear discrepancy of a partially ordered set P=(X,), denoted by ld(P), is the least integer k for which there exists an injection f:XZ satisfying (i) if xy, then f(x)<f(y); and (ii) if x and y are incomparable, then |f(x)–f(y)|k. There is an apparent connection between ld(P) and the bandwidth of the incomparability graph of P. We prove that, in fact, these two quantities are always equal.