A class of non-Desargusian planes

A new class of non-Desargusian planes of order q², where q is a power of an odd prime, is constructed. These planes have the interesting property that they all admit a collineation group of order (q² ? 1).     

The class of projective planes is noncomputable

Computable projective planes are investigated. It is stated that a free projective plane of countable rank in some inessential expansion is unbounded. This implies that such a plane has infinite computable dimension. The class of all computable projective planes is proved to be noncomputable (up to computable isomorphism).     

A new class of square order planes

Only a few classes of square order planes are known. These are generalized André planes (including Hall planes), flag transitive planes, Hering's planes and Walker's planes. A new class of planes of order 5^2r, where r is an odd natural number, is constructed and the translation complements of the corresponding planes have been determined. The translation complements divide the sets of distinguished points into 4 orbits of lengths 1, 1, 5^r ? 1, and 5^2r ? 5^r and it is of order 5^r(5^r ? 1)² if r ≠ 1. In the particular case of r = 1, the translation complement divides the set of distinguished points into two orbits of lengths 6 and 20 and it is of order 480.     

Projective planes of Lenz�CBarlotti class V

We overview some recent results on projective planes of Lenz?CBarlotti class V.     

A class of d-dimensional flag transitive translation planes

Journal of Geometry -     

On a class of partial planes related to biplanes

In this note we consider partial planes in which for each element x (point or line) there exists a unique opposite element or antipode x* which cannot be joined to x or has no intersection with x. We also require the existence of a triangle. Such partial planes will be called antipodal planes. We are mainly interested in the subclass of regular antipodal planes satisfying: p I L implies p* I L* for all points p and lines L. We shall provide a free construction of infinite regular antipodal planes. The objects thus constructed are not free objects in the usual sense since between antipodal planes there do not exist proper homomorphisms. On the other hand, regular antipodal planes do have a canonical homomorphic image which is a biplane (cf. Payne, J Comb Theory A 12:268–282, 1972). Regular antipodal planes can be coordinatized by certain algebraic systems in a similar way as projective planes are coordinatized by ternary rings. Again by a free construction, we shall provide examples satisfying a configuration theorem comparable to the Fano condition with fixed line at infinity.     

A class of fibered loops related to general hyperbolic planes

In this paper we introduce a class of left conjugacy closed loops which are also fibered in subsemigroups. We inspect the possibility to extend the semigroups of the fibration to commutative subgroups. Then we construct an example of such loops arising from a suitable selected subset of the set of all limit rotations of the hyperbolic plane over an euclidean field.