Collineations of finite 2-affine planes

We investigate collineations of finite 2-affine planes and show that with the exception of some 2-affine planes of order at most 4 each collineation is induced by a collineation of its projective completion. We further deal in more detail with 2-affine planes of type II and their collineations that fix each long line.     

ϰ-affine Räume

The -affine space is defined, i.e. a geometry in which distinct points are joint by exactly one curve (subspace of dimension 1). This definition generalizes the notions of affine [1] and Möbius [4] spaces. An example of a 5-affine space with the Mathieu group as automorphism group is constructed. There are only a few models of at least 4-affine planes of finite order.     

Bericht über κ-affine Geometrien

Ohne ZusammenfassungMeinem verehrten Lehrer, Herrn Prof. Dr. HELMUT KARZEL in Dankbarkeit gewidmet.Diesem Bericht liegt meine Habilitationsschrift -affine RÄume zugrunde. Auszugsweise habe ich über den Gegenstand dieser Arbeit auf der Conference on Geometry in Waterloo/Ont. 1971 vorgetragen.     

2-affine complete algebras need not be affine complete

For each we exhibit a finite algebra R _k such that R _k is k-affine complete, but not (k+1)-affine complete; this means that every k-ary congruence preserving function on R _k lies in , but there is a (k +1)-ary congruence preserving function of R _k that does not lie in . Received September 27, 2001; accepted in final form February 9, 2002.     

Chambers of 2-affine arrangements and freeness of 3-arrangements

We give a lower bound on the number of chambers divided by a family of lines in the real plane. Also, a sufficient condition for 3-arrangements to be free is given in terms of chambers. To prove them, we give the upper bound on differences of exponents for balanced 2-multiarrangements in terms of the cardinality of hyperplanes.     

Dual local completeness

The 1971 articles in which Saxon-Levin and Valdivia independently proved their Theorem feature two conditions equivalent to dual local completeness. One became Ruess' property (LC). The other is among new characterizations previously known only as necessary conditions.

     

On the structure of indecomposable injective modules 1

For a ring R, the following two conditions are equivalent:.

(1) If E is an indecomposable injective right R-module, then End E_R is a field (not necesarily commutative).

(2) Every co-irreducible rigtht ideal is critical.

Since (2) has been characterized ideal-theoretically, this amounts to an ideal-theoretical characterization of (1). These rings come up to the study of (QI) rings in which every quasi-injective module is injective.     

Completions of certain modules over quantum groups 1

Completion is a procedure of constructing new representations for Lie algebras from given ones. This paper shows that a similar procedure exists for quantum groups.     

Local completeness I

An infinite algebra is locally complete if its local closure is the set of all finitary operations. We give a local completeness criterion in terms of a system R of finitary relations on A such that the polymorph Pol of each R is locally incomplete and for every locally incomplete algebra A; F we have F Pol for some R. This system consists of (i) certain natural relations whose polymorphs are best possible in the sense that they are co-atoms in the lattice of locally closed incomplete algebras, (ii) five types of binary relations, (iii) one type of ternary relations and (iv) at least ternary totally reflexive and symmetric relations that are not locally central.Presented by K. A. Baker.     

Model completeness for trivial, uncountably categorical theories of Morley rank 1

We show that if T is a trivial uncountably categorical theory of Morley Rank 1 then T is model complete after naming constants for a model.The second author was partially supported by NSF grant DMS-0300080.     

On N1-free modules with trivial dual

In this paper we generalize the notion of Fountain-Gould order to the context of associative pairs. We give theorems about the existence and uniqueness of the left quotient pair and we obtain a Goldie-like haracterization which describes left orders in semiprime pairs coinciding with their socle.     

Varieties of modules and affine modules

Acta Mathematica Hungarica -     

Crossed modules and doi-hopf modules

We prove that crossed modules (or Yetter-Drinfel’d modules) are special cases of Doi’s unified Hopf modules. The category of crossedH-modules is therefore a Grothendieck category (if we work over a field), and the Drinfel’d double appears as a type of generalized smash product. The second and the third author both thank the University of Brussels for its warm hospitality during their visit there. This author was supported partially by the CNCSU nr. 221 and by the Romanian academy, grant nr. 3721.