Regular semigroups with weakly simplistic orthodox transversals

Weakly simplistic orthodox transversals are introduced in this paper and some characterizations associated with them are obtained. An example is given to demonstrate that weakly simplistic orthodox transversals are proper generalisations of left simplistic orthodox transversals. The related results of Blyth and Almeida Santos on left simplistic inverse transversals obtained in 1996 and Kong and Luo on left simplistic orthodox transversals obtained in 2011 are generalised and amplified. A structure theorem of regular semigroups with weakly simplistic orthodox transversals is also established.     

具有左简纯正断面的正则半群

本文引入并研究了左简纯正断面, 得到了与之相关的若干刻画; 推广并丰富了Blyth 和AlmeidaSantos 于1996 年得到的关于左简逆断面及Kong 于2007 年得到的关于纯正断面的相关结果; 同时,给出了具有左简纯正断面的正则半群的结构定理.     

On Generalized Orthodox Transversals

Generalized orthodox transversals are introduced in this article, and some characterizations associated with them are obtained. The related results of Chen on quasi-ideal orthodox transversals obtained in 1999 and Zhang and Wang on generalized inverse transversals obtained in 2008 are generalized and amplified. A structure theorem of a regular semigroup with a perfect generalized orthodox transversal is also established.     

具有Clifford断面的正则纯正半群

给出了具有Clifford断面的右正规纯正半群的等价刻画,得到了具有Clifford断面的正则纯正半群的次直积分解,证明了具有Clifford断面的正则纯正半群一定是正则纯正群．     

Free orthodox semigroups and free bands with inverse transversals

It is well known that the subclass of inverse semigroups and the subclass of completely regular semigroups of the class of regular semigroups form the so called e-varieties of semigroups. However, the class of regular semigroups with inverse transversals does not belong to this variety. We now call this class of semigroups the ist-variety of semigroups, and denote it by IST. In this paper, we consider the class of orthodox semigroups with inverse transversals, which is a special ist-variety and is denoted by OIST. Some previous results given by Tang and Wang on this topic are extended. In particular, the structure of free bands with inverse transversals is investigated. Results of McAlister, McFadden, Blyth and Saito on semigroups with inverse transversals are hence generalized and enriched.     

具有某种断面的半群的研究进展

本文综述了几类具有特殊断面的半群的近期研究结果。在介绍逆半群和正则半群的一般结构之后，概述了具有逆断面的正则半群的结构和同余格的研究成果。总结了作为逆断面的推广的可裂断面，纯正断面，正则^*－断面和恰当断面。提出了可以进一步研究的重要的问题。     

HOPs and COPs: Room frames with partitionable transversals

In this paper, we construct Room frames with partitionable transversals. Direct and recursive constructions are used to find sets of disjoint complete ordered partitionable (COP) transversals and sets of disjoint holey ordered partitionable (HOP) transversals for Room frames. Our main results include upper and lower bounds on the number of disjoint COP transversals and the number of disjoint HOP transversals for Room frames of type 2ⁿ. This work is motivated by the large number of applications of these designs.     

Quasi-ideal transversals of abundant semigroups and spined products

We provide a new and much simpler structure for quasi-ideal adequate transversals of abundant semigroups in terms of spined products, which is similar in nature to that given by Saito in Proc. 8th Symposium on Semigroups, pp. 22–25 (1985) for weakly multiplicative inverse transversals of regular semigroups. As a consequence we deduce a similar result for multiplicative transversals of abundant semigroups and also consider the case when the semigroups are in fact regular and provide some new structure theorems for inverse transversals.     

具有逆断面的纯正群的构造

本文用具有半格断面的带和Clifford半群给出具有逆断面的纯正群的一个构造定理.     

带有乘右适当断面的右主投射半群(英文)

In this paper,the concept of right adequate transversals of rpp semigroups is introduced.We establish the structure of rpp semigroups with multiplicative right adequate transversals in terms of right normal bands and right adequate semigroups.In particular, some special cases are considered.     

On the Multiplicative Quasi-Adequate Transversals of an Abundant Semigroup

In this article, we explore the multiplicative quasi-adequate transversals of an abundant semigroup. Let S be an abundant semigroup with multiplicative quasi-adequate transversals. Then the product of any two multiplicative quasi-adequate transversals is also a multiplicative quasi-adequate transversal. Moreover, all multiplicative quasi-adequate transversals of S form a rectangular band. Let S°and S ^? be multiplicative quasi-adequate transversals of S, and let δ°(δ^?) be the δ-relation on S°(S ^?). Then there exists a bijection ? from S°/δ°onto S ^?/δ^?. In particular, if δ°and δ^? are congruences, then the bijection ? is an isomorphism.     

On Generalized Adequate Transversals

The aim of this article is to study abundant semigroups with generalized adequate transversals. We obtain some properties of such semigroups and give, in particular, a construction of the class of abundant semigroups with quasi-ideal generalized adequate transversals. Finally, we apply this construction to some special cases.     

Parallel focal structure and singular Riemannian foliations

We give a necessary and sufficient condition for a submanifold with parallel focal structure to give rise to a global foliation of the ambient space by parallel and focal manifolds. We show that this is a singular Riemannian foliation with complete orthogonal transversals. For this object we construct an action on the transversals that generalizes the Weyl group action for polar actions.

     

On the combinatorics of plethysm

We construct three (large, reduced) incidence algebras whose semigroups of multiplicative functions, under convolution, are anti-isomorphic, respectively, to the semigroups of what we call partitional, permutational and exponential formal power series without constant term, in infinitely many variables x = (x₁, x₂,…), under plethysm. We compute the Möbius function in each case. These three incidence algebras are the linear duals of incidence bialgebras arising, respectively, from the classes of transversals of partitions (with an order that we define), partitions compatible with permutations (with the usual refinement order), and linear transversals of linear partitions (with the order induced by that on transversals). We define notions of morphisms between partitions, permutations and linear partitions, respectively, whose kernels are defined to be, in each case, transversals, compatible partitions and linear transversals. We introduce, in each case, a pair of sequences of polynomials in x of binomial type, counting morphisms and monomorphisms, and obtain expressions for their connection constants, by summation and Möbius inversion over the corresponding posets of kernels.     

Good congruences on abundant semigroups with PSQ-adequate transversals

In this paper, we will characterize good congruences on abundant semigroups with \(PSQ\) -adequate transversals by using the so called good congruence triple, which generalizes some results on abundant semigroups with multiplicative adequate transversals.     

Transversals to Line Segments in Three-Dimensional Space

We completely describe the structure of the connected components of transversals to a collection of n line segments in ℝ³. Generically, the set of transversals to four segments consists of zero or two lines. We catalog the non-generic cases and show that n≥ 3 arbitrary line segments in ℝ³ admit at most n connected components of line transversals, and that this bound can be achieved in certain configurations when the segments are coplanar, or they all lie on a hyperboloid of one sheet. This implies a tight upper bound of n on the number of geometric permutations of line segments in ℝ³.     

Good congruences on abundant semigroups with multiplicative quasi-adequate transversals

In this paper, some properties of good homomorphic images of abundant semigroups with a multiplicative quasi-adequate transversal are obtained. After that, good congruences on a class of abundant semigroups with such transversals are studied. We characterize the good congruences by using the so-called good congruence triple. Furthermore, we prove that any good congruence on a class of abundant semigroups with such transversals can be constructed by the good congruence triple.     

On the Helly Number for Hyperplane Transversals to Unit Balls

We prove two results about the Hadwiger problem of finding the Helly number for line transversals of disjoint unit disks in the plane, and about its higher-dimensional generalization to hyperplane transversals of unit balls in d -dimensional Euclidean space. These consist of (a) a proof of the fact that the Helly number remains 5 even for arbitrarily large sets of disjoint unit disks—thus correcting a 40-year-old error; and (b) a lower bound of d+3 on the Helly number for hyperplane transversals to suitably separated families of unit balls in R ^d . Received January 25, 1999, and in revised form July 7, 1999.     

A Helly-type theorem for higher-dimensional transversals

We prove that a collection of compact convex sets of bounded diameters in that is unbounded in k independent directions has a k-flat transversal for k<d if and only if every d+1 of the sets have a k-transversal. This result generalizes a theorem of Hadwiger(–Danzer–Grünbaum–Klee) on line transversals for an unbounded family of compact convex sets. It is the first Helly-type theorem known for transversals of dimension between 1 and d−1.     

Lopsided Lovász Local Lemma and Latin transversals

A new version of the Lovász Local lemma is used to prove the existence of Latin transversals in matrices where no symbol appears too often.