格蕴涵N序半群中的sl理想

研究由格蕴涵代数所诱导的格蕴涵N序半群的sl理想的代数特性，给出了格蕴涵代数L中的LI-理想与相应的格蕴涵N序半群中的sl理想以及理想之间的关系定理，并对由集合A所生成的sl理想进行了刻画。最后，讨论了L中的准素sl理想和素sl理想之间的关系。     

完全分配格与点格

本文第一部分利用完备格上的上拓扑子基,给出完全分配格与点格的若干新刻划,并讨论其上的 Scott 拓扑与 Lawson 拓扑的基与子基的构造.第二部分讨论点格与代数格的关系,证明了 L 是点格当且仅当 L 为代数格且 L~(op)为完全 Heyting代数,并证明了代数偏序集范畴与点格范畴是等价的.     

幂等元分配半环簇的格 *

给出了自由幂等元分配半环上的字问题的解 ,借助于这个解 ,确定了幂等元分配半环簇ID的子簇格L(ID)∶L(ID)同构于格L和一个四元格的直积 ,其中L表示带簇B的子簇格L(B)的4个拷贝的次直积 ;L(ID)是可数无限的和分配的 ;ID的每一个子簇有有限基底.     

格蕴涵代数的LI-理想格及其素元刻画

运用格理论的原理和方法对格蕴涵代数L的LI-理想概念作进一步研究.首先,在L的全体LI-理想之集ф_(LI)(L)上定义了格运算■和■,蕴涵运算■以及伪补运算■,证明了(ф_(LI)(L),,■,■,■,{O},L)构成一个完备Heyting代数的结论.其次,利用运算固的性质给出了(ф_(LI)(L),,■,■,■,■,{O},L)成为Boolean代数的若干充要条件.最后,借助于L的素LI-理想之特性获得了格(ф_(LI)(L),,■,■,{O},L)中素元的若干等价刻画.     

格矩阵半群的Euler-Fermat公式

给出并证明格矩阵半群的Euler-Fermat公式:A(n-1)2 1 = A(n-1)2 1 [n], A ∈ Mn(L)其中L是任意的分配格,Mn(L)是L上所有n阶矩阵构成的半群.这是布尔矩阵半群的Euler-Fermat公式的一种推广.     

内射拓扑分子格

本文证明了任给T_O拓扑分子格(L,η),以下三条等价:(1)(L,η)为正则内射拓扑分子格;(2)L为完备集环且其完备余素元集ht(L)形成一连续格,余拓扑η为该连续格ht(L)上的Scott闭集格;(3)存在T_O内射拓扑空间(X,Τ),(L,η)同胚于(P(X),Τ~c)在拓扑分子格范畴中的Sober化。此外,还给出了正则内射拓扑分子格、(一般)内射拓扑分子格以及正则内射分子格的一般结构。作为应用,重新证明了有指数元的拓扑分子格的结构。     

半连续格的半素极小集与序同态扩张

对完备格引入半素极小集的概念,证明完备格L为半连续格当且仅当L中的每个元在L中存在半素极小集,给出半连续格的两个序同态扩张定理.     

格蕴涵代数定义的简化

证明定义有二元运算→和零元运算(固定元)0的非空集合L(即:L是(2,0)型代数)只要满足四条算律就可成为格蕴涵代数.因此,在定义格蕴涵代数时,我们不必要求L是有界有余格,从L是(2,0)型代数出发即可,这样大大简化了格蕴涵代数的定义.     

广义λ超连续格的关系表示

给出广义λ超连续格的几个刻画。特别地,我们证明了完备格L上的λ-区间拓扑θλ(L)是严格T2的L是广义λ超连续格L上的关系是广义λ正则的。     

格点形心问题的若干结果

设n(k)为满足如下条件的最小整数，给定平面上任意n个格点，其中必存在k个点的形心也是格点，文献[4]提出关于确定n(4)的未解问题，本文给出解答n(4)=13，并进一步给出相关的一些问题的结果。     

On a Characterization Theorem of Semimodular Lattices

In general lattice theory,there are two basic criterion theorems:　 (1 ) A lattice L is modular if and only if L does not contain a pentagon N5(seeFig. 1 a) .　 (2 ) A lattice L is distributive ifand only if L contains neither a pentagon N5nor a dia-mond M3(see Fig. 1 b) .　 Now,the problem is that if L is semimodular,whether L possesses the similar char-acteristic?In this regard,the following conclusion is obtained in this paper:　 A lattice L is semimodular if and only if L does not co…     

Extra two-fold Steiner pentagon systems

A two-fold pentagon system is a decomposition of the complete 2-multigraph (every two distinct vertices joined by two edges) into pentagons. A two-fold Steiner pentagon system is a two-fold pentagon system such that every pair of distinct vertices is joined by a path of length two in exactly two pentagons of the system. We consider two-fold Steiner pentagon systems with an additional property : for any two vertices, the two paths of length two joining them are distinct. We determine completely the spectrum for such systems, and point out an application of such systems to certain 4-cycle systems.     

On affine images of regular pentagons and pentagrams

In this paper we explore pentagons that are affine images of the regular pentagon and the regular pentagram. We obtain their characterizations in terms of two mild forms of regularity that deal with the notions of medians for a pentagon and the natural requirement that they are concurrent. Using these characterizations we show that there are various values involving the number 5 (thus related to the golden section) for which a careful selection of division points on appropriate segments determined by any pentagon will result in a pentagon that is the affine image of either a regular pentagon or a regular pentagram.     

Steiner pentagon systems

A Steiner pentagon system is a pair (K_n, P) where K_n isthe complete undirected graph on n vertices. P is a collection of edge-disjoint pentagons which partition K_n, and such that every part of distinct vertices of K_n is joined by a path of length two in exactly one pentagon of the collection P. The number n is called the order of the system. This paper gives a somplete solution of the existence problem of Steiner pentagon systems. In particular it is shown that the spectrum for Steiner pentagon systems (=the set of all orders for which a Steiner pentagon system exists) is precisely the set of all n ≡ 1 or 5 (mod 10), except 15, for which no such system exists.     

Orthogonal triangulation of polygons

In this study, the question whether any convex polygon can be divided using an orthogonal grid into right-angled triangles is answered in the negative. Moreover, it is demonstrated that there exists a convex pentagon which cannot be even approximated by divisible pentagons.     

Symmetry and the golden ratio in the analysis of a regular pentagon

The regular pentagon had a symbolic meaning in the Pythagorean and Platonic philosophies and a subsequent important role in Western thought, appearing also in arts and architecture. A property of regular pentagons, which was probably discovered by the Pythagoreans, is that the ratio between the diagonal and the side of these pentagons is equal to the golden ratio. Here, we will study some relations existing between a regular pentagon and this ratio. First, we will focus on the group of fivefold rotational symmetry, to find the position in the complex plane of the vertices of this geometric figure. Then, we will propose an analytic method to solve the same problem based on the Cartesian coordinates, a method where we find the golden ratio without any specific geometric consideration. This study shows a comparison of the use of complex numbers, symmetries and analytic methods, applied to a subject which can be interesting for general education in mathematics. In fact, the proposed approach can convey and link several concepts, requiring only a general pre-college education, showing at the same time the richness that mathematics can offer in solving geometric problems.     

Intervals in lattices of quasiorders

We investigate the structure of intervals in the lattice of all closed quasiorders on a compact or discrete space. As a first step, we show that if the intervalI has no infinite chains then the underlying space may be assumed to be finite, and in particular,I must be finite, too. We compute several upper bounds for its size in terms of its heighth, which in turn can be computed easily by means of the least and the greatest element ofI. The cover degreec of the interval (i.e. the maximal number of atoms in a subinterval) is less than 4h. Moreover, ifc4(n–1) thenI contains a Boolean subinterval of size 2 ⁿ , and ifI is geometric then it is already a finite Boolean lattice. While every finite distributive lattice is isomorphic to some interval of quasiorders, we show that a nondistributive finite interval of quasiorders is neither a vertical sum nor a horizontal sum of two lattices, with exception of the pentagon. Many further lattices are excluded from the class of intervals of quasiorders by the fact that no join-irreducible element of such an interval can have two incomparable join-irreducible complements. Up to isomorphism, we determine all quasiorder intervals with less than 9 elements and all quasiorder intervals with two complementary atoms or coatoms.     

Super-Simple Twofold Steiner Pentagon Systems

A twofold pentagon system of order v is a decomposition of the complete undirected 2-multigraph 2K _v into pentagons. A twofold Steiner pentagon system of order v [TSPS(v)] is a twofold pentagon system such that every pair of distinct vertices is joined by a path of length two in exactly two pentagons of the system. A TSPS(v) is said to be super-simple if its underlying (v, 5, 4)-BIBD is super-simple; that is, if any two blocks of the BIBD intersect in at most two points. In this paper, it is shown that the necessary conditions for the existence of a super-simple TSPS(v); namely, v ≥ 15 and v ≡ 0 or 1 (mod 5) are sufficient. For these specified orders, the main result of this paper also guarantees the existence of a very special and interesting class of twofold and fourfold Steiner pentagon systems of order v with the additional property that, for any two vertices, the two or four paths of length two joining them are distinct.     

Convex pentagons that admit <Emphasis Type="Italic">i</Emphasis>-block transitive tilings

The problem of classifying the convex pentagons that admit tilings of the plane is a long-standing unsolved problem. Previous to this article, there were 14 known distinct kinds of convex pentagons that admit tilings of the plane. Five of these types admit tile-transitive tilings (i.e. there is a single transitivity class with respect to the symmetry group of the tiling). The remaining 9 types do not admit tile-transitive tilings, but do admit either 2-block transitive tilings or 3-block transitive tilings; these are tilings comprised of clusters of 2 or 3 pentagons such that these clusters form tile-2-transitive or tile-3-transitive tilings. In this article, we present some combinatorial results concerning pentagons that admit i-block transitive tilings for \(i \in \mathbb {N}\). These results form the basis for an automated approach to finding all pentagons that admit i-block transitive tilings for each \(i \in \mathbb {N}\). We will present the methods of this algorithm and the results of the computer searches so far, which includes a complete classification of all pentagons admitting i-block transitive tilings for \(i \le 4\), among which is a new 15th type of convex pentagon that admits a tile-3-transitive tiling.     

A counterexample to the generalization of Sperner's theorem

It has been conjectured that the analog of Sperner's theorem on non-comparable subsets of a set holds for arbitrary geometric lattices, namely, that the maximal number of non-comparable elements in a finite geometric lattice is max w(k), where w(k) is the number of elements of rank k. It is shown in this note that the conjecture is not true in general. A class of geometric lattices, each of which is a bond lattice of a finite graph, is constructed in which the conjecture fails to hold.