The Δ<Superscript>0</Superscript><Subscript><Emphasis Type="Italic">α</Emphasis></Subscript>-Computable Enumerations of the Classes of Projective Planes

Studying computable representations of projective planes, for the classes K of pappian, desarguesian, and all projective planes, we prove that K ^c /? admits no hyperarithmetical Friedberg enumeration and admits a Friedberg Δ⁰_α+3-computable enumeration up to a Δ⁰ _α -computable isomorphism.     

Continuous planar functions

This paper deals with continuous planar functions and their associated topological affine and projective planes. These associated (affine and projective) planes are the so-called shift planes and in addition to these, in the case of planar partition functions, the underlying (affine and projective) translation planes. We introduce a method that allows us to combine two continuous planar functions ? → ? into a continuous planar function ?² → ?². We prove various extension and embedding results for the associated affine and projective planes and their collineation groups. Furthermore, we construct topological ovals and various kinds of polarities in the associated topological projective planes.     

Planar Functions and Planes of Lenz-Barlotti Class II

Planar functions were introduced by Dembowski and Ostrom [4] to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a question posed in [4]. These planar functions define at least one such affine plane of order 3^e for every e 4 and their projective closures are of Lenz-Barlotti type II. All previously known planes of type II are obtained by derivation or lifting. At least when e is odd, the planes described here cannot be obtained in this manner.     

A Semifield Plane of Odd Order Admitting an Autotopism Subgroup Isomorphic to <Emphasis Type="Italic">A</Emphasis><Subscript>5</Subscript>

We develop an approach to constructing and classifying semifield projective planes with the use of a spread set. The famous conjecture is discussed on the solvability of the full collineation group of a finite semifield nondesarguesian plane. We construct a matrix representation of a spread set of a semifield plane of odd order admitting an autotopism subgroup isomorphic to the alternating group A₅ and find a series of semifield planes of odd order not admitting A₅.     

Ovoids and translation planes revisited

Kantor has previously described the translation planes which may be obtained by projecting sections of ovoids in ⁺(8, q)-spaces to ovoids in corresponding ⁺(6, q)-spaces. Since the Klein correspondence associates spreads in 4-dimensional vector spaces with ovoids in ⁺(6, q)-spaces, there are corresponding translation planes of order q ² and kernel containing GF(q). In this article, we revisit some of these translation planes and give some presentations of the spreads. Motivated by various properties of the planes, we study, in general, translation planes which admit certain homology groups and/or elation groups. In particular, we develop new constructions of projective planes of Lenz-Barlotti class II-1.Finally, we show how certain projective planes of order q ² of Lenz-Barlotti class II-1 may be considered equivalent to flocks of quadratic cones in PG(3, q).This work was partially supported by NSF grant DMS-8800843.     

Two-transitive groups on a hyperbolic unital

A classification given previously of all projective translation planes of order q² that admit a collineation group G admitting a two-transitive orbit of q+1 points is applied to show that the only projective translation planes of order q² admitting a hyperbolic unital acting two-transitively on a secant are the Desarguesian planes and the unital is a Buekenhout hyperbolic unital.     

Smooth Hughes planes are classical

We prove that the only compact projective Hughes planes which are smooth projective planes are the classical planes over the complex numbers \Bbb C \Bbb C , the quaternions \Bbb H \Bbb H , and the Caley numbers \Bbb O \Bbb O . As a by-product this shows that an 8-dimensional smooth projective plane which admits a collineation group of dimension d 3 17d \geq 17 is isomorphic to the quaternion projective plane P ₂\Bbb H {\cal P _2\Bbb H }. For topological compact projective planes this is true if d 3 19d \geq 19, and this bound is sharp.     

A simplicial 3-arrangement of 22 planes

This paper describes a join-irreducible simplicial arrangement A _inf1 ^sup3 (22) of 22 planes in real projective 3-space that is not included in the list recently published by Branko Grünbaum and G. C. Shephard. Seventeen sporadic 3-arrangements are now known.     

Topologische Divisionsalgebren ohne zugehörige topologische affine Ebene

We prove that many (non-associative) topological division algebrasD of dimensionn ∈ N over the centreK do not yield topological affine or projective planes (of Lenz-Barlotti type V) in contrast to the results of SKORNJAKOV [20], SALZMANN [18] and [19], GRUNDHÖFER [7], HARTMANN [11] and RINK [17] concerning projective planes coordinatized by compact or special topological ternary fields. In particular, this holds for every non-trivial and non-archimedian valuation topology ofK distinct from the order topology ifK is a real-closed field, and if the division algebraD =K ⁿ carries the product topology.     

The Projective Class Ring of Basic and Split Hopf Algebras

We describe the structure of the complexification of the projective class ring of a basic and split Hopf algebra using a positive integer determined by the composition series of the projective cover of the trivial module. If q is a root of unity of order n, the projective class algebra of u_q ⁺(sl₂) is the product of n–1 copies of the dual numbers over C and two copies of C.     

On the projective differential geometry of a non-holonome surface in ordinary space

Sunto. The paper deals with many projective properties of the neighbourhoods of the third,4th, 5th order of a point on a non-holonome surfaceV ₃ ² inS ₃. Chiefly two remarkable projective correspondences are studied, between the tangent plane and the bundle of directions, with his contact point as centre. Many generalizations are obtained of geometrical loci (so as lines, planes, quadrics) projectively associated with the neighbourhoods of a point on an ordinary (holonome) surface. The last sections are concerned with the extension to theV ₃ ² of the quadric ofMoutard and theSegre's correspondence.      

A note on the projective differential geometry of a non-holonome surface

Sunto. The paper is concerned with projective differential properties of the plane sections made by the planes which pass through a non-asymptotic tangent, of a non-holonome surfaceV ₃ ² inS ₃. Chiefly by this way some surfaces are determined, which are projectively connected with the generic point on theV ₃ ² .      

Einbettung gewisser Kettengeometrien in projektive Räume

The theory of “chain geometries” as represented in [2] is a generalisation of the concept of Möbius-, Laguerre- and pseudo-euclidean planes over a commutative field K. It is well known that these geometries can be represented as a 2-dimensional variety of the 3-dimensional projective space over K. It will be shown how to embed in a similar way a class of “chain geometries”, which covers these planes. The algebras belonging to these geometries are the kinematic algebras, studied by H.KARZEL, in which x²? Kx+K for each element x of the algebra. If the algebra is of rank n the geometry will be represented on a n-dimensional algebraic variety of the (n+1)-dimensional projective space π, the chains being the intersection of with planes of π having no line but at least two points in common with .     

Semifield planes of odd order that admit a subgroup of autotopisms isomorphic to <Emphasis Type="Italic">A</Emphasis> <Subscript>4</Subscript>

We develop an approach to constructing and classification of semifield projective planes with the use of a linear space and a spread set. We construct a matrix representation of the spread set of a semifield plane of odd order that admits a Baer involution in the translation complement or a subgroup of autotopisms isomorphic to the alternating group A₄. We give examples of semifield planes of order 81 satisfying the above indicated condition.     

Projective Equivalence of Δ-Matroids with Coefficients and Symplectic Geometries

The projective equivalence of matroid representations over fields and of oriented matroids is well studied. This paper is devoted to the study of projective equivalence of Δ-matroids with coefficients, which covers the concept of projective equivalence of matroids with coefficients and thus in particular the projective equivalence of represented and oriented matroids. A necessary and sufficient condition for the projective equivalence of Δ-matroids with coefficients is established in terms of the inner Tutte group T _M ⁽⁰⁾ of the underlying combinatorial geometry M. The structure of T_M ⁽⁰⁾ of symplectic projective spaces M of odd dimensions d≥3 over fields is computed.     

Cartesian groups not belonging to topological projective planes

A new class of topological Cartesian groups is introduced by bending the lines of the affine coordinate plane over an ordered skewfield countably infinitely often. These Cartesian groups belong to topological affine planes with continuous parallelism, but they need not yield a topological projective plane if the topology is not equal to the order topology of the skewfield, in contrast to the results of HARTMANN [6] for generalized Moulton planes. Dedicated to Professor Dr. Gerhard Grimeisen on his 70 ^th birthday     

A class of unitals of order q which can be embedded in two different planes of order q2

By deriving the desarguesian plane of order q² for every prime power q a unital of order q is constructed which can be embedded in both the Hall plane and the dual of the Hall plane of order q² which are non-isomorphic projective planes. The representation of translation planes in the fourdimensional projective space of J. André and F. Buekenhouts construction of unitals in these planes are used. It is shown that the full automorphism groups of these unitals are just the collineation groups inherited from the classical unitals.     

Switching Sets Regolari E Cubiche Rigate DiP G(5,q) Piani Di Tralsazione Ad Essi AssociatiG(5,q) Piani Di Tralsazione Ad Essi Associati

In a recent paper, the authors studied some algebraic hypersurfaces of the third order in the projective spacePG(5,q) and they called them ruled cubics, since they possess three systems of planes. Any two of these constitute a regular switching set and furthermore, if Σ is a given regular spread ofPG(5,q), one of the three systems is contained in Σ. The subject of this note is to prove, conversely, that every regular switching set (Φ, Φ′) with Φ ? Σ is a ruled cubic and to construct, for a generic choice of the projective reference system inP G(5,q), the quasifield which coordinatizes the translation plane Π associated with the spread (Σ ? Φ) ∪ Φ′. The planes Π, of orderq ³, are a generalization of the finite Hall planes.     

Constructing cotorsion pairs over generalized path algebras

We introduce two adjoint pairs (e _i ^λ , ( ) _i ) and (( ) _i , e _i ^ρ ) and give a new method to construct cotorsion pairs. As applications, we characterize all projective and injective representations of a generalized path algebra and exhibit projective and injective objects of the category M _p which is a generalization of monomorphisms category.