Planar Nearrings Research Articles

Overlaps in field generated circular planar nearrings

We investigate circular planar nearrings con-structed from finite fields as well the complex number field using amultiplicative subgroup of order k, and characterize the overlaps of the basic graphs which arise in the associated 2-designs.

On improving the construction of balanced incomplete block designs using planar near-rings

The purpose of this paper is to highlight the importance of planar near-rings (PNR) in the construction of balanced incomplete block designs (BIBD), then to give a new precise formula to blocks of this type of BIBD which enables us to generate useful matrices based on the incidence matrix. Moreover, we will improve the efficiency factor and determine the condition to produce a symmetric BIBD from a BIBD.

Distribution and generalized centre in planar nearrings

Nearrings are the nonlinear generalization of rings. Planar nearrings play an important role in nearring theory, both from the structural side, being close to generalized nearfields, as well as from an applications perspective, in geometry and combinatorial designs related to difference families. In this paper we investigate the distributive elements of planar nearrings. If a planar nearring has nonzero distributive elements, then it is an extension of an abelian group by its zero multiplier part. In the case that there are distributive elements that are not zero multipliers, then this extension splits, giving an explicit description of the nearring, a coordinatisation result. This generalizes the structure of planar rings. We provide a family of examples where this does not occur, the distributive elements being precisely the zero multipliers. We apply this knowledge to the question of determining the generalized centre of planar nearrings as well as finding new proofs of older results.

BIBDs algorithm

In this paper, we explore the planar near-rings to build BIBDs (Balanced Incomplete Block Designs) which is an important tool in the field of the theory of error-correcting code. Our purpose is to supply a program which determines BIBDs and their incidence matrices from a finite field, what gives us the possibility of choosing the best BIBD among the BIBDs obtained according to the values of its parameters.

Planar nearrings on the Euclidean plane

Planar near-rings are generalized rings which can serve as coordinate domains for geometric structures in which each pair of nonparallel lines has a unique point of intersection. It is known that all planar nearrings can be constructed from regular groups of automorphisms of groups which can be viewed as the “action groups” of the planar nearring. In this article, we study planar nearrings whose additive group is \({(\mathbb{R}^n,+)}\), in particular, n = 1 and 2. It is natural to study topological planar nearrings in this context, following ideas of the late Kenneth D. Magill, Jr. In the case of n = 1, we characterize all topological planar nearrings by their action groups \({(\mathbb{R}^*, \cdot)}\) or \({(\mathbb{R}^+, \cdot)}\). For n = 2, these action groups and the circle group \({(\mathbb{U}, \cdot)}\) seem to be the most interesting cases, but the last case can be excluded completely. As a consequence, we obtain characterizations of the semi-homogeneous continuous mappings from \({\mathbb{R}^n}\) to \({\mathbb{R}}\) for n = 1 and 2. Such a mapping f enjoys the property that f(f(u)v) = f(u)f(v) for all \({u,v \in \mathbb{R}^n}\). When \({f(\mathbb{R}^n) = \mathbb{R}^+}\), f is a positive homogeneous mapping of degree 1.

Planar Near-Rings And Coding Theory

The purpose of this paper is to discuss the importance of algebraic coding theory and to investigate the special case in which BIB designs and codes are constructed from planar near-rings. Application of planar near-rings to binary codes were first explored by Modisett [12] and by Fuchs, Hofer and Pilz [14]. Key word: planar near-ring, incidence structure, tactical configuration, BIBD, binary codes, block code.

FROM PLANAR NEARRINGS TO GENERATING BLOCKS

Planar nearrings, like most of the subjects in finite geometries, can be applied for the construction of block designs. In this article we introduce the ideas of constructing simple BIBDs (balanced incomplete block designs) from field-generated (or nearfield-generated) planar nearrings. Further investigation reveals that there are strong connections between these kinds of constructions and the action of a sharply 2-transitive group on a set. We next explore the structures of the constructions. The theory is derived from finite fields directly. The main point is on finding a subset $S$ (called generating block) of the field $F$ with respect to the given stabilizer $Stab_{F^*}(S)$. A big portion of simple BIBDs with various parameters can be obtained in this way; many simple BIBDs with the same parameters appear. We classify the constructed BIBDs according to the types of the respective generating blocks.

Abstract Algebra in Statistics

In this note we show how some specific classes of algebraic structures (“planar nearrings”)give rise to efficient Balanced Incomplete Block Designs, which in turn can excellently beused in statistical experiments.

Matrix maps over planar near-rings

Following a method by Meldrum and van der Walt, near-rings of matrix maps are defined for general near-rings, not necessarily with identity. The influence of one-sided identities is discussed. When the base near-ring is integral and planar, the near-ring of matrix maps is shown to be simple. Various types of primitivity of the near-ring of matrix maps are discussed when the base near-ring is planar but not integral. Finally, an open problem concerning bijective matrix maps is solved.

FROM PLANAR NEARRINGS TO GENERATING BLOCKS

Planar nearrings, like most of the subjects in finite geometries, can be applied for the construction of block designs. In this article we introduce the ideas of constructing simple BIBDs (balanced incomplete block designs) from field-generated (or nearfield-generated) planar nearrings. Further investigation reveals that there are strong connections between these kinds of constructions and the action of a sharply 2-transitive group on a set. We next explore the structures of the constructions. The theory is derived from finite fields directly. The main point is on finding a subset S (called generating block) of the field F with respect to the given stabilizer StabF∗ (S). A big portion of simple BIBDs with various parameters can be obtained in this way; many simple BIBDs with the same parameters appear. We classify the constructed BIBDs according to the types of the respective generating blocks.

Partially balanced incomplete block designs from weakly divisible nearrings

In [ [6] Riv. Mat. Univ. Parma 11 (2) (1970) 79–96] Ferrero demonstrates a connection between a restricted class of planar nearrings and balanced incomplete block designs. In this paper, bearing in mind the links between planar nearrings and weakly divisible nearrings (wd-nearrings), first we show the construction of a family of partially balanced incomplete block designs from a special class of wd-nearrings; consequently, we are able to give some formulas for calculating the design parameters.

On Finite Circular Planar Nearrings

The main subject of this paper is to characterize finite circular Ferrero pairs, the tool for constructing finite circular planar nearrings. A general characterization of finite circular Ferrero pairs is given. In particular, a generalized version of Modisett's characterization (see J. R. Clay “Nearrings: Geneses and Applications,” pp. 68–75, Oxford Univ. Press, Oxford, 1992) is presented for finite circular Ferrero pairs (N,Φ) with cyclic Φ. We show that the fixed point free group of automorphisms Φ of a finite circular Ferrero pair (N,Φ) is metacyclic. Finally, two questions on the existence of nonabelian circular planar nearrings are answered by a general construction method and some basic examples.

Planar Rings

In this note we characterize all planar near-rings that are rings.

Strongly monogenic near-rings

Introduction. A (left) zero-symmetric near-ring N is said to be strongly monogenic (s.m.) if a N s {0, N} for each a ~ N and b N = N for some b ~ N. It is known that every planar near-ring has this property. We have already studied such near-rings [5 and 6] in connection with the non-commutative geometry in the sense of Andr6 (see e.g. [1]): here we want to consider them in their own right. In his book [4], Pilz asked to find the radicals of planar near-rings. These (prime, nil and Jacobson-type radicals) have been recently found by Gallina [2, Corollary 20, p. 15] for a class of s.m. near-rings, containing all the planar ones. The problem, therefore, has been solved. In the first section we generalize this result, by showing that the description of the radicals given in [2] does hold for every s.m. near-ring (Theorem 1.7). We see there that such a N has an unique maximal ideal R, which turns out to be equal to all the said

Cocyclic planar near-rings

Cocyclic planar near-rings

Planar and strongly uniform near-rings

The concept of a finite strongly uniform near-ring defined by G. Ferrero is generalized to the infinite case. A relation between planar near-rings and strongly uniform near-rings is studied. A structure theorem for an integral planar near-ring of M. Anshel and J. Clay is extended to a strongly uniform nearring.