拟连续格的逆极限

给出以保任意交和定向并为态射的拟连续格范畴的逆极限,讨论函子保广义拟连续格逆极限的条件。     

Global-Affine Morphisms of Projective Lattice Geometries

The fundamental theorem of projective geometry gives an algebraic representation of isomorphisms between projective geometries of dimension at least 3 over vector spaces and has been generalized in different ways. This note briefly presents some further generalizations which will be proved in the author’s thesis. We introduce the notion of global-affine morphisms between projective lattice geometries. Our investigations result in a general partial representation of global-affine morphisms which yields a complete representation of global-affine homomorphisms between large classes of module-induced projective geometries by semilinear mappings between the underlying modules.     

On the category of weak Cayley table morphisms between groups

Weak Cayley table functions between groups are generalized conjugacy-preserving homomorphisms, under which products of images are conjugate to images of products. There is a weak Cayley table bijection between two groups iff they have the same 2-characters. In this paper, weak Cayley table functions are augmented to include the specific conjugating elements, leading to the concept of a weak (Cayley table) morphism. If the conjugating elements are chosen subject to a crossed-product condition, then the weak morphisms between groups form a category. The forgetful functor to this category from the category of group homomorphisms is shown to possess a left adjoint. Two weak morphisms are said to be homotopic if they project to the same weak Cayley table function. As a first step in the analysis of the category of weak morphisms, the group of units of the monoid of weak morphisms homotopic to the identity automorphism of a group is described.     

Categorical Aspects in Projective Geometry

After introducing morphisms between projective geometries, some categorical questions are examined. It is shown that there are three kinds of embeddings and two kinds of quotients. Furthermore the morphisms decompose in a canonical way into four factors.     

Reichhaltige Intervalle in Untergruppenverbänden

A natural notion of morphisms between projective geometries is introduced. The classical correspondence of projective geometries with certain lattices is extended into an equivalence of categories. Furthermore, quotient geometries are defined. The associated projections are shown to be those morphisms having a right inverse, the inclusion of subspaces those having a left inverse.Supported by a grant from the Fonds national suisse de la recherche scientifique.     

Automorphisms of Mn,partially ordered by the star order

Let Mn be the space of all n × n matrices with coefficients in [image omitted] or [image omitted], where n ≥ 3. The star order on Mn is defined by [image omitted] iff [image omitted], where A* is the Hermitian adjoint (i.e., the conjugate transpose) of A. We characterize surjective mappings Φ on Mn such that [image omitted] iff [image omitted]. The tools we use are the Fundamental theorem of projective geometry, Wigner's theorem, and the Penrose decomposition, which we need to describe the main result as well.     

TWO CLASSES OF HOMOMORPHISMS IN MODULE CATEGORIES

Abstract

In a category R-Mod a homomorphism α:A → B is called projective if α factors through every epimorphism with B as image. Injective homomorphisms are defined dually. Some properties of such homomorphisms are derived, and it is shown that the hereditariness of the ring R is equivalent to some conditions which can be simply stated in terms of projective and injective homomorphisms.     

Differential projective modules over differential rings

Differential modules over a commutative differential ring which are projective as ring modules, with differential homomorphisms, form an additive category. Every projective ring module is shown occurs as the underlying module of a differential module. Differential modules, projective as ring modules, are shown to be direct summands of differential modules free as ring modules; those which are differential direct summands of differential direct sums of the ring being induced from the subring of constants. Every differential module finitely generated and projective as a ring module is shown to have this form after a faithfully flat finitely presented differential extension of the base.     

Projective representations II. generalized chain geometries

In this paper, projective representations of generalized chain geometries are investigated, using the concepts and results of [5]. In particular, we study under which conditions such a projective representation maps the chains of a generalized chain geometry Σ (F, R) to reguli; this mainly depends on how the field F is embedded in the ring R. Moreover, we determine all bijective morphisms of a certain class of generalized chain geometries with the help of projective representations. Dedicated to Walter Benz on the occasion of his 70th birthday. The first author was supported by a Lise Meitner Research Fellowship of the Austrian Science Fund (FWF), projects M529-MAT, M574-MAT..     

Morphisms of projective geometries and semilinear maps

Morphisms between projective geometries are introduced; they are partially defined maps satisfying natural geometric conditions. It is shown that in the arguesian case the morphisms are exactly those maps which in terms of homogeneous coordinates are described by semilinear maps. If one restricts the considerations to automorphisms (collineations) one recovers the so-called fundamental theorem of projective geometry, cf. Theorem 2.26 in [2].Supported by a grant from the Fonds National Suisse de la Recherche Scientifique.     

Zur Hyperebenenalgebraisierung in desargues-Schen projektiven Verbandsgeometrien

As a completion and extension of a result of A. Day and D. Pickering [5] we obtain the following structure theorem in the conceptual frame of projective lattice geometries: In a Desarguesian projective geometry the subgeometry of every at least one-dimensional hyperplane is module induced.

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Ich danke Dr. S. E. Schmidt ganz herzlich für alle kritischen und wertvollen Anregungen während der Entstehung dieser Arbeit.

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Prof. Armin Herzer zum 60. Geburtstag gewidmet     

投射可迁环上矩阵环的若当同态

设R′是一个环,Mn′（R′）是R′上的n′×n′矩阵环.如果环R有不变基数性质并且每个有限生成的投射左R-模是自由模,则R是一个投射自由环.如果环R≌Mr（S）,其中S是一个投射自由环,则R是一个投射可迁环.当R是一个投射可迁环时,给出了从Mn′（R′）到Mn（R）（n′≥n≥2）的若当同态的代数公式.     

Projective geometries over rings and modular lattices

In this paper, by analogy with the classical case, projective geometries over rings are constructed and connections between them and projective lattices are studied. Also, morphisms between ring geometries and projective lattices are described. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 46, Algebra, 2007.     

Embeddings and hyperplanes of the Lie incidence geometry of type E7,1

This paper continues a program to show that for most of the standard Lie incidence geometries, all geometric hyperplanes arise from a necessarily absolutely universal embedding, by addingE _7,1 to the list. It follows from [5, 12] that any projective embedding of this point line geometry is a homomorphic image of the one afforded by the 56-dimensional module for the groupE ₇(K).This work was supported by a grant from the National Science Foundation.     

Finite geometries which contain dual affine planes

Finite geometries in which each plane is projective or dual affine over the field of two elements, or affine over the field of three elements, are studied. It is shown that no connected geometry can mix all three species of planes, and the geometries in which projective and dual affine planes occur are classified.     

RP—环

本文定义了ＲＰ－环,讨论了它的性质和等价条件,得出了Ｓ＝Ｍｎ（Ｒ）是ＲＰ－环妥且仅当Ｒ是ＲＰ－环;Ｒｉ是ＲＰ－环当且仅当每个Ｒｉ是ＲＰ－环．最后讨论了ＲＰ－环的同态象和左投射维数．     

New Maximal Arcs in Desarguesian Planes

Constructions are described of maximal arcs in Desarguesian projective planes utilizing sets of conics on a common nucleus in PG(2, q). Several new infinite families of maximal arcs in PG(2, q) are presented and a complete enumeration is carried out for Desarguesian planes of order 16, 32, and 64. For each arc we list the order of its stabilizer and the numbers of subarcs it contains. Maximal arcs may be used to construct interesting new partial geometries, 2-weight codes, and resolvable Steiner 2-designs.     

The design of linear algebra and geometry

Conventional formulations of linear algebra do not do justice to the fundamental concepts of meet, join, and duality in projective geometry. This defect is corrected by introducing Clifford algebra into the foundations of linear algebra. There is a natural extension of linear transformations on a vector space to the associated Clifford algebra with a simple projective interpretation. This opens up new possibilities for coordinate-free computations in linear algebra. For example, the Jordan form for a linear transformation is shown to be equivalent to a canonical factorization of the unit pseudoscalar. This approach also reveals deep relations between the structure of the linear geometries, from projective to metrical, and the structure of Clifford algebras. This is apparent in a new relation between additive and multiplicative forms for intervals in the cross-ratio. Also, various factorizations of Clifford algebras into Clifford algebras of lower dimension are shown to have projective interpretations.As an important application with many uses in physics as well as in mathematics, the various representations of the conformal group in Clifford algebra are worked out in great detail. A new primitive generator of the conformal group is identified.