Units of compatible nearrings

We study singly-generated wavelet systems on ${\mathbb {R}^2}$ that are naturally associated with rank-one wavelet systems on the Heisenberg group N. We prove a necessary condition on the generator in order that any such system be a Parseval frame. Given a suitable subset I of the dual of N, we give an explicit construction for Parseval frame wavelets that are associated with I. We say that ${g\in L^2(I\times \mathbb {R})}$ is Gabor field over I if, for a.e. ${\lambda \in I}$ , |??|^1/2 g(??, ·) is the Gabor generator of a Parseval frame for ${L^2(\mathbb {R})}$ , and that I is a Heisenberg wavelet set if every Gabor field over I is a Parseval frame (mother-)wavelet for ${L^2(\mathbb {R}^2)}$ . We then show that I is a Heisenberg wavelet set if and only if I is both translation congruent with a subset of the unit interval and dilation congruent with the Shannon set.     

Units of compatible nearrings,IV

This is the fourth in a sequence of papers originating in a effort to study the units of a compatible nearring $R$ satisfying the descending chain condition on right ideals using a faithful compatible module $G$ of $R$ . A key point in this endeavor involves determining $1 + Ann_R(G/H)$ where $H$ is a direct sum of isomorphic minimal $R$ -ideals where success in doing so gives us not only information about the units of $R$ , but also information about $R$ and $J_2(R)$ . In the previous papers, $1 + Ann_R(G/H)$ has been determined whenever $G/H$ does not contain a minimal factor isomorphic to the minimal summands of $H$ . In this paper we determine $1 + Ann_R(G/H)$ when $G/H$ does contain a minimal factor isomorphic to the minimal summands of $H$ . With the completion of the determination of $1 + Ann_R(G/H)$ in all cases, we illustrate how things work in practice by considering the nearrings generated by the inner automorphisms of a finite dihedral group, special linear group, and general linear group and nearrings of congruence preserving functions on an expanded group.     

Units of compatible nearrings, III

This paper is a continuation of two previous works studying the units of a compatible nearring R satisfying the descending chain condition on right ideals using a faithful compatible module G of R. A crucial point in doing this involves determining 1 +  Ann _R (G/H) where H is a direct sum of isomorphic minimal R-ideals. The high point of this paper is extending this determination from the cases in the previous works to the case where G/H and H contain no isomorphic minimal factors. We also shall further expand our knowledge of when a special type of principal series for G introduced in the second of these previous works called a quasi c-chain exists.     

Compatible extensions of nearrings

If R is a zero-symmetric nearring with 1 and G is a faithful R-module, a compatible extension of R is a subnearring S of M ₀(G) containing R such that G is a compatible S-module and the R-ideals and S-ideals of G coincide. The set of these compatible extensions forms a complete lattice and we shall study this lattice. We also will obtain results involving the least element of this lattice related to centralizers and the largest element of this lattice related to uniqueness of minimal factors with an application to 1-affine completeness of the R-module G.     

Compatible extensions of nearrings

If R is a zero-symmetric nearring with 1 and G is a faithful R-module, a compatible extension of R is a subnearring S of M ₀(G) containing R such that G is a compatible S-module and the R-ideals and S-ideals of G coincide. The set of these compatible extensions forms a complete lattice and we shall study this lattice. We also will obtain results involving the least element of this lattice related to centralizers and the largest element of this lattice related to uniqueness of minimal factors with an application to 1-affine completeness of the R-module G.     

New designs from circular nearrings

Nearrings are generalized rings in which addition is not in general abelian and only one distributive law holds. Some interesting combinatorial structures, as tactical configurations and balanced incomplete block designs (BIBDs) naturally arise when considering the class of planar and circular nearrings. We define the concept of disk and prove that in the case of field-generated planar circular nearrings it yields a BIBD. Such designs can be used in the construction of some classes of codes for which we are able to calculate the parameters.     

Projective Klingenberg planes over nearrings

In the first section the definition of a PK-nearring is given and it is shown that every PK-nearring is a local ternary ring. Then a PK-nearring is used to construct PK-planes and further algebraic properties of PK-nearrings are related to geometrical ones of the corresponding PK-planes. Finally, a wide class of PK-nearrings is presented by applying the Dickson-process to the ring of formal power series.Throughout the following all nearrings shall be left-distributive. For the notation and basic properties of nearrings the reader is referred to Pilz [5].Dedicated to Professor Helmut Karzel on the occasion of his 65. birthday     

Left zero covering homomorphisms of laminated nearrings

Nearrings here are right nearrings. LetN be a nearring and fix an element α εN. Form another nearring N_α by taking addition to be the same as the addition inN but define the productx ⊗y of two elementsx, y ε N_α byx ⊗y =xay. The nearring N_α is referred to as a laminated nearring ofN andN is referred to as the base nearring. The element α is called the laminating element or the laminator. An elementx of a nearingN is a left zero ifxy =x for ally εN. A homomorphismϕ from a nearringN ₁ into a nearringN ₂ is a left zero covering homomorphism if for each left zeroy εN ₂,ϕ(x) =y for somex εN ₁. The left zero covering homomorphisms from one laminated nearring into another are investigated where the base nearring is the nearring of all continuous selfmaps of the Euclidean group ℝ² under pointwise addition and composition and the laminators are complex polynomials. Finally, it is shown that one can determine whether or not two such laminated nearrings are isomorphic simply by inspecting the coefficients of the two laminating polynomials.     

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.     

Local nearrings with generalized quaternion multiplicative group

Abstract A not necessarily zero-symmetric nearring R with identity is called local if the set of all non-invertible elements of R forms a subgroup of its additive group. The local nearrings whose multiplicative group is generalized quaternion are described. In particular, it is proved that their additive groups are abelian of types (3,3), (2,2,2,2), (2,2,4), (2,2,2,2,2) and (2,2,2,4). Keywords: Local nearring, Multiplicative group, Generalized quaternion group, Factorized group Mathematics Subject Classification (2000): 16N20, 16U60, 20M25     

Topological nearrings whose additive groups are Euclidean

Let (G,+) be a group with a locally compact Hausdorff topology for which the binary operation + is continuous. Those, binary operation * onG for which (G, +, *) is a topological nearring are described. In the case whereG is abelian, those binary operations * for which (G, +, *) is a topological ring are also described. Versions of these results are then obtained in the special case where the group is the topological Euclideann-group,R ⁿ. A family of binary operations * for which (R ⁿ, +, *)_is a topological nearring is then investigated in some detail. Most of these nearrings turn out to be planar. Their ideals are completely determined and we characterize those nearrings which are simple. The multiplicative semi-groups (R ⁿ, *) of these nearrings are then investigated. Green's relations are completely determined and it is shown that a number of familiar properties of semigroups are equivalent for these particular semigroups. Finally, all those binary operations * for which (R, +, *) is a topological nearring are completely described. It is determined when any two of these nearrings are isomorphic and for each of these nearrings, its automorphism group, is completely determined.     

PBIBDs from weakly divisible nearrings and related codes

In [5] the authors are able to give a method for the construction of a family of partially balanced incomplete block designs from a special class of wd-nearrings (wd-designs). In this paper the wd-design incidence matrix and the connected row and column codes are studied. The parameters of two special classes of wd-designs and those of the related row and column codes are calculated.     

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.     

Stability of compatible schedules

The change of an optimal structural schedule with the inclusion of a new activity is studied. For an activity of a unit volume, an algorithm of rearrangement of a schedule is developed and an estimate of its complexity is obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 21–34, 1983.     

The size of 3-compatible, weakly compatible split systems

A split system on a finite set X is a set of bipartitions of X. Weakly compatible and k-compatible (k??1) split systems are split systems which satisfy special restrictions on all subsets of a certain fixed size. They arise in various areas of applied mathematics such as phylogenetics and multi-commodity flow theory. In this note, we show that the number of splits in a 3-compatible, weakly compatible split system on a set X of size n is linear in?n.