On Unitals ith Many Baer Sublines

We identify the points of PG(2, q) ith the directions of lines in GF(q ³), viewed as a 3-dimensional affine space over GF(q). Within this frameork we associate to a unital in PG(2, q) a certain polynomial in to variables, and show that the combinatorial properties of the unital force certain restrictions on the coefficients of this polynomial. In particular, if q = p ² where p is prime then e show that a unital is classical if and only if at least (q - 2) secant lines meet it in the points of a Baer subline.     

Unitals Admitting All Translations

Among all 2‐‐designs, we characterize the Hermitian unitals by the existence of sufficiently many translations. In arbitrary 2‐‐designs, each group of translations with given center acts semiregularly on the set of points different from the center.     

Unitals and Unitary Polarities in Symmetric Designs

We extend the notion of unital as well as unitary polarity from finite projective planes to arbitrary symmetric designs. The existence of unitals in several families of symmetric designs has been proved. It is shown that if a unital in a point-hyperplane design PG _d-1(d,q) exists, then d = 2 or 3; in particular, unitals and ovoids are equivalent in case d = 3. Moreover, unitals have been found in two designs having the same parameters as the PG ₄(5,2), although the latter does not have a unital. It had been not known whether or not a nonclassical design exists, which has a unitary polarity. Fortunately, we have discovered a unitary polarity in a symmetric 2-(45,12,3) design. To a certain extent this example seems to be exceptional for designs with these parameters.     

Unitals in the code of the Hughes plane

A coding‐theoretic characterization of a unital in the Hughes plane is provided, based on and extending the work of Blokhuis, Brouwer, and Wilbrink in PG(2,q²). It is shown that a Frobenius‐invariant unital is contained in the p‐code of the Hughes plane if and only if that unital is projectively equivalent to the Rosati unital. © 2003 Wiley Periodicals, Inc.     

Computing Linear Codes and Unitals

The 2-rank of any 2-(28,4,1) design (unital on 28 points) is known to be between 19 and 27. It is shown by the enumeration and analysis of certain binary linear codes that there are no unitals of 2-rank 20, and that there are exactly 4 isomorphism classes of unitals of 2-rank 21. Combined with previous results, this completes the classification of unitals on 28 points of 2-rank less than 22.     

Characterizing the Hermitian and Ree Unitals on 28 Points

We prove a conjecture of Brouwer, namely that a 2-(28,4,1) design has 2–rank at least 19, with equality occuring if and only if the design is the Ree unital. We give a similar characterization of the Hermitian unital.     

Self-Orthogonal Compact Spreads and Unitals in Topological Translation Planes

A spread of is a set of l-dimensional subspaces L V partitioning V {0}. We construct examples of compact spreads that are identical with their sets of orthogonal spaces L . In the corresponding topological translation planes, every Euclidean sphere is a unital with the additional property that every point at infinity has flat feet.     

Unitals in Finite Desarguesian Planes

We show that a suitable 2-dimensional linear system of Hermitian curves of PG(2,q²) defines a model for the Desarguesian plane PG(2,q). Using this model we give the following group-theoretic characterization of the classical unitals. A unital in PG(2,q²) is classical if and only if it is fixed by a linear collineation group of order 6(q + 1)² that fixes no point or line in PG(2,q²).     

Unitals and inversive planes

We show that if U is a Buekenhout-Metz unital (with respect to a point P) in any translation plane of order q ² with kernel containing GF(q), then U has an associated 2-(q²,q+1,q) design which is the point-residual of an inversive plane, generalizing results of Wilbrink, Baker and Ebert. Further, our proof gives a natural, geometric isomorphism between the resulting inversive plane and the (egglike) inversive plane arising from the ovoid involved in the construction of the Buekenhout-Metz unital. We apply our results to investigate some parallel classes and partitions of the set of blocks of any Buekenhout-Metz unital.     

Unitals in the Hall plane

The unitals in the Hall plane are studied by deriving PG(2,q ²)and observing the effect on the unitals of PG(2,q ²).The number of Buekenhout and Buekenhout-Metz unitals in the Hall plane is determined. As a corollary we show that the classical unital is not embeddble in the Hall plane as a Buekenhout unital and that the Buekenhout unitals of H(q ²)are not embeddable as Buekenhout unitals in the Desarguesian plane. Finally, we generalize this technique to other translation planes.     

Subregular Spreads of Hermitian Unitals

In this paper, we consider the problem of constructing partitions of the points of a Hermitian unital into pairwise disjoint blocks, commonly known as spreads. We generalize a construction of Baker et al. (In Finite Geometry and Combinatorics, Vol. 191 of London Math. Soc. Lecture Not Ser., pages 17–30. Cambridge University Press, Cambridge, 1993.) to provide a new infinite family of spreads. Morover, we develop a structural connection between these new spreads of the Hermitian unital in PG(2, q²) and the subregular spreads of PG(3, q), allowing us to christen a new “subregular” family of spreads in the Hermitian unital in PG(2, q²).     

Unitals in the regular nearfield planes

Classes of parabolic unitals in the regular nearfield planes of odd square order are enumerated and classified. These unitals correspond to certain Buekenhout-Metz unitals in the classical plane. Their collineation groups are determined and the unitals are sorted by projective equivalence.      

Approximately unital operator systems

The main theme of this paper is to consider a notion of 'approximately unital operator systems' including both C*-algebras and unital operator systems.The goals are to prove a version of the Choi-Effros theorem for these systems,to introduce a functorial process for forming an approximately unital operator systems from a given matrix ordered vector space with a proper approximate order unit,to study second duals of these objects and to prove that a C*-algebra can be characterized as an approximately unital ...     

Albert algebras over curves of genus zero and one

Albert algebras and other Jordan algebras are constructed over curves of genus zero and one, using a generalization of the Tits process and the first Tits construction due to Achhammer.     

Presentations of subgroups of the braid group generated by powers of band generators

According to the Tits conjecture proved by Crisp and Paris (2001) [4], the subgroups of the braid group generated by proper powers of the Artin elements σi are presented by the commutators of generators which are powers of commuting elements. Hence they are naturally presented as right-angled Artin groups.The case of subgroups generated by powers of the band generators aij is more involved. We show that the groups are right-angled Artin groups again, if all generators are proper powers with exponent at least 3. We also give a presentation in cases at the other extreme, when all generators occur with exponent 1 or 2. Such presentations are distinctively more complicated than those of right-angled Artin groups.     

Some New Upper Bounds for the Size of Partial Ovoids in Slim Generalized Polygons and Generalized Hexagon of Order (s, s3)

From an elementary observation, we derive some upper bounds for the number of mutually opposite points in the classical generalized polygons having 3 points on each line. In particular, it follows that the Ree-Tits generalized octagon O(2) of order (2, 4) has no ovoids. Also, we deduce from another observation a similar upper bound in any generalized hexagon of order (s, s ³).     

On the dimensions of the binary codes of a class of unitals

Let U_β be the special Buekenhout-Metz unital in PG(2,q²), formed by a union of q conics, where q=p^e is an odd prime power. It can be shown that the dimension of the binary code of the corresponding unital design U_β is less than or equal to q³+1−q. Baker and Wantz conjectured that equality holds. We prove that the aforementioned dimension is greater than or equal to .     

Collineation groups of translation planes admitting hyperbolic Buekenhout or parabolic Buekenhout-Metz unitals

It is shown that for every semifield spread in PG(3,q) and for every parabolic Buekenhout-Metz unital, there is a collineation group of the associated translation plane that acts transitively and regularly on the affine points of the parabolic unital. Conversely, any spread admitting such a group is shown to be a semifield spread. For hyperbolic Buekenhout unitals, various collineation groups of translation planes admitting such unitals and the associated planes are determined.     

The Change-Base Issue for Ω-Categories

Let G ： Ω→Ω＇ be a closed unital map between commutative, unital quantales. G induces a functor G^- from the category of Ω-categories to that of Ω＇-categories. This paper is concerned with some basic properties of G^-. The main results are： （1） when Ω, Ω＇ are integral, G ： Ω→Ω＇ and F ： Ω＇→Ω are closed unital maps, F is a left adjoint of G^- if and only if F is a left adjoint of G; （2） G^- is an equivalence of categories if and only if G is an isomorphism in the category of commutative unital quantales and closed unital maps; and （3） a sufficient condition is obtained for G^- to preserve completeness in the sense that GA is a complete Ω＇-category whenever A is a complete Ω-category.