Lexicographic Products of Half Linearly Ordered Groups

The notion of the half linearly ordered group (and, more generally, of the half lattice ordered group) was introduced by Giraudet and Lucas [2]. In the present paper we define the lexicographic product of half linearly ordered groups. This definition includes as a particular case the lexicographic product of linearly ordered groups. We investigate the problem of the existence of isomorphic refinements of two lexicographic product decompositions of a half linearly ordered group. The analogous problem for linearly ordered groups was dealt with by Maltsev [5]; his result was generalized by Fuchs [1] and the author [3]. The isomorphic refinements of small direct product decompositions of half lattice ordered groups were studied in [4].     

On half cyclically ordered groups

In this paper we introduce and investigate the notion of half cyclically ordered group generalizing the notion of half partially ordered group whose study was begun by Giraudet and Lucas.     

Affine completeness and wreath product decompositions of lattice ordered group

Let Δ and H be a nonzero abelian linearly ordered group or a nonzero abelian lattice ordered group, respectively. In this paper we prove that the wreath product of Δ and H fails to be affine complete. Supported by VEGA grant 2/4134/24. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information, grant I/2/2005.     

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.     

Affine Completeness and Lexicographic Product Decompositions of Abelian Lattice Ordered Groups

In this paper it is proved that an abelian lattice ordered group which can be expressed as a nontrivial lexicographic product is never affine complete. This work was supported by VEGA grant 2/1131/21.     

Small cycle double covers of products I: Lexicographic product with paths and cycles

Bondy conjectured that every simple bridgeless graph has a small cycle double cover (SCDC). We show that this is the case for the lexicographic products of certain graphs and along the way for the Cartesian product as well. Specifically, if G does not have an isolated vertex then G □ P₂ and G □ C_2k have SCDCs. If G has an SCDC then so does G □ P_k, k > 2 and G □ C_{2k + 1}. We use these Cartesian results to show that P_2j[G] (j ≥ 1) and C_k[G] (k ≠ 3, 5, 7) have SCDCs. Also, if G has an SCDC then so does P_{2j + 1}[G] (j ≥ 4). The results for the lexicographic product are harder and, in addition to the Cartesian results, require certain decompositions of K_n,n into perfect matchings. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 99–123, 2008     

图的字典积的非零整数流(英文)

充分利用图的字典积的结构证明了以下结论:如果图G_1的每连通分支都非平凡,图G_2的阶数大于3,那么它们的字典积G_1[G_2]具有非零3-流.     

有向图字典乘积的代数群性质

字典乘积有向图G_1→⊙G_2是通过已知阶数较小的有向图G_1和G_2构造来的,这些小有向图G_1和G_2的拓扑结构和性质肯定影响大有向图G_1→⊙G_2的拓扑结构和性质.运用群论方法,证明了有向图字典乘积的一些代数性质,如:结合律、分配律等.     

Temporal logic of linearly ordered <Emphasis Type="Italic">α</Emphasis>-spaces

Using the language of temporal logic, we construct a decidable calculus L* α and prove that the calculus is complete w.r.t. the class of all strictly linearly ordered α-frames. Supported by RFBR (project No. 06-01-00358) and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-335.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 723–749, November–December, 2008.     

On the Jump Number of Lexicographic Sums of Ordered Sets

Let Q be the lexicographic sum of finite ordered sets Q _x over a finite ordered set P. For some P we can give a formula for the jump number of Q in terms of the jump numbers of Q _x and P, that is, , where s(X) denotes the jump number of an ordered set X. We first show that where w(X) denotes the width of an ordered set X. Consequently, if P is a Dilworth ordered set, that is, s(P) = w(P)–1, then the formula holds. We also show that it holds again if P is bipartite. Finally, we prove that the lexicographic sum of certain jump-critical ordered sets is also jump-critical.     

图的笛卡儿积及字典式积的连通性

文章给出了两个图的笛卡儿积及字典式的积为最大边连通的、最大连通的、super-λ,super-κ及hyper-κ的充分条件,同时证明了其中一些条件也是必要的.此外,对这两种积的局部割集和广义割集的性质也进行了考虑.     

An analog of the fundamental theorem of arithmetic in ordered groupoids

In the note we consider ordered groupoids with the Riesz interpolation property, that is, ifa _i ≤b _j (i, j=1,2), then there exists ac such thata _i ≤c≤b _j (i, j=1,2). For such groupoids possessing the descending chain condition for the positive cone and the property

A non-metrizable compact linearly ordered topological space, every subspace of which has a -minimal base

A collection of subsets of a space is minimal if each element of contains a point which is not contained in any other element of . A base of a topological space is -minimal if it can be written as a union of countably many minimal collections. We will construct a compact linearly ordered space satisfying that is not metrizable and every subspace of has a -minimal base for its relative topology. This answers a problem of Bennett and Lutzer in the negative.

     

广义字典序积的自同态幺半群

讨论了图的广义字典序积的自同态幺半群的性质,给出了广义字典序积图X[Yz|x∈V（X）]的自同态幺半群与X,Yx(x∈V(X))的自同态幺半群的圈积相等的充要条件。     

On linearly ordered <Emphasis Type="Italic">H</Emphasis>-closed topological semilattices

We give a criterion for a linearly ordered topological semilattice to be H-closed. We also prove that any linearly ordered H-closed topological semilattice is absolutely H-closed and we show that every linearly ordered semilattice is a dense subsemilattice of an H-closed topological semilattice.     

Hamilton cycle decompositions of the tensor product of complete multipartite graphs

In this paper, tensor product of two regular complete multipartite graphs is shown to be Hamilton cycle decomposable. Using this result, it is immediate that the tensor product of two complete graphs with at least three vertices is Hamilton cycle decomposable thereby providing an alternate proof of this fact.     

Resolvable even cycle decompositions of the tensor product of complete graphs

In this paper, we consider resolvable k-cycle decompositions (for short, k-RCD) of K_m×K_n, where × denotes the tensor product of graphs. It has been proved that the standard necessary conditions for the existence of a k-RCD of K_m×K_n are sufficient when k is even.     

FI代数同构于一族全序FI代数的直积的子代数的条件

解决了王国俊教授提出的关于FI代数的嵌入问题,并进一步研究了FI代数与一族全序FI代数的直积的子代数同构的条件.主要结果是:交换FI代数可同构嵌入一族全序交换FI代数的直积;分配FI代数可同构嵌入一族全序分配FI代数的直积.     

Schauder decompositions and the Fremlin projective tensor product of Banach lattices

In this paper, we first introduce a lattice decomposition and finite-dimensional lattice decomposition (FDLD) for Banach lattices. Then we show that for a Banach lattice with FDLD, the following are equivalent: (i) it has the Radon-Nikodym property; (ii) it is a KB-space; (iii) it is a Levi space; and (iv) it is a σ-Levi space. We then give a sequential representation of the Fremlin projective tensor product of an atomic Banach lattice with a Banach lattice. Using this sequential representation, we show that if one of the Banach lattices X and Y is atomic, then the Fremlin projective tensor product has the Radon-Nikodym property (or, respectively, is a KB-space) if and only if both X and Y have the Radon-Nikodym property (or, respectively, are KB-spaces).