EQUIPRIME Γ-NEAR-RINGS

Abstract

Equiprime and strongly equiprime near-rings were recently defined by the present authors, together with S. Veldsman. In the present paper, the concepts are introduced for Γ-near-rings, and give rise to Kurosh- Amitsur radicals. If M is a Γ-near-ring and L is its left operator near-ring, then R(L)⁺ = R(M), where R(—) in both cases denotes either the equiprime or the strongly equiprime radical.     

Profinitely generated Γ-modules

Theorems of homological definiteness of arbitrary -modules are proved which generalize similar results proved earlier for -modules of special type.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 86, pp. 157–161, 1979.     

A novel approach towards fuzzy Γ-ideals in ordered Γ-semigroups

In many applied disciplines like computer science, coding theory and formal languages, the use of fuzzified algebraic structures especially ordered semigroups play a remarkable role. In this paper, we introduce a new concept of fuzzy Γ-ideal of an ordered Γ-semigroup G called an (∈, ∈ ?q _k )-fuzzy Γ-ideal of G. Fuzzy Γ-ideal of type (∈, ∈ ∨q _k ) are the generalization of ordinary fuzzy Γ-ideals of an ordered Γ-semigroup G. A new characterization of ordered Γ-semigroups in terms of an (∈, ∈ ∨q _k )-fuzzy Γ-ideal is given. We show that a fuzzy subset λ of an ordered Γ-semigroup G is an (∈, ∈ ∨q _k )-fuzzy Γ-ideal of G if and only if U (λ; t) is a Γ-ideal of G for all \(t \in \left( {0,\frac{{1 - k}} {2}} \right]\) . We also investigate some important characterization theorems in terms of this notion. Finally, regular ordered Γ-semigroups are characterized by the properties of their (∈, ∈ ∨q _k )-fuzzy Γ-ideals.     

The characterization of Γ-modules in terms of fuzzy soft Γ-submodules

In this paper, we focus on combining the theories of fuzzy soft sets with Γ-modules, and establishing a new framework for fuzzy soft Γ-submodules. The main contributions of the paper are 3-fold. First, we present the concepts of (R, S)-bi-Γ-submodules, quasi-Γ-submodules and regular Γ-modules. Meanwhile, some illustrative examples are given to show the rationality of the definitions introduced in this paper. Second, several new kinds of generalized fuzzy soft Γ-submodules are proposed, and related properties and mutual relationships are also investigated. Third, we discover some intrinsic connections between the generalized fuzzy soft Γ-submodules presented in this paper and crisp Γ-submodules, and describe the relationships between regular Γ-modules and the generalized fuzzy soft Γ-submodules presented in this paper.     

λ-invariants and Γ-transforms

In this paper we give the relationship between the -invariants of two power series which are naturally associated to the -transform of ap-adic measure. Since thep-adicL-functions over arise as such -transforms, we obtain information about the minus part of Iwasawa's -invariant for the basic _p -extension of an abelian CM-field.I would like to thank W. Sinnott, R. Gold and T. Dowling for their comments and suggestions.     

Homological determination of Γ-modules

It is proved that if M is a profinitely generated -module which is free as a module over the ring of p-adic integers, then M is determined up to free direct factors by its homology. This result generalizes the theorem on homological determinacy of p-adic representations of a cyclic group [Ref. Zh. Mat., 3A, 318 (1971)].Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeieniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 104–126, 1976.     

Wiener's criterion and Γ-convergence

Dirichlet problems with homogeneous boundary conditions in (possibly irregular) domains and stationary Schrödinger equations with (possibly singular) nonnegative potentials are considered as special cases of more general equations of the form –u + µu = 0, whereµ is an arbitrary given nonnegative Borel measure in ⁿ . The stability and compactness of weak solutions under suitable variational perturbations ofµ is investigated and stable pointwise estimates for the modulus of continuity and the energy of local solutions are obtained.     

Limiting angles of Γ-martingales

Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m≥ 3 and that, outside a fixed compact set, the sectional curvatures are bounded above by −c ₁/{r ² ln r} and below by −c ₂ r ², where c ₁ and c ₂ are two positive constants and r is the geodesic distance from a fixed point. We show that, when κ≥ 1 satisfies certain conditions, the angular part of a κ-quasi-conformal Γ-martingale on M tends to a limit as time tends to infinity and the closure of the support of the distribution of this limit is the entire sphere at infinity. This improves both a result of Le for Brownian motion and also results concerning the non-existence of κ-quasi-conformal harmonic maps from certain types of Riemannian manifolds into M. Received: 19 September 1997     

RADICALS IN GENERAL Γ-NEAR-RINGS

Abstract

The J ₂ and J ₃ radicals for zerosymmetric Γ-near-rings were recently defined by the author. In the present paper we define the J ₂₍₀₎ and J ₃₍₀₎ radicals for arbitrary Γ-near-rings. These radicals are sirmlar to corresponding ones which were recently defined by Veldsman for near-rings. Let M be a r-near-ring with left operator near-ring L. Then J _κ(0)(L)⁺ = J _{κ (0)} (M), k. = 2,3. If A is an ideal of M, then J _{κ (0)} (A) ? J _{κ (o)}(M) ∩ A, with equality when k = 3 and A is left invariant. J ₃₍₀₎ is a Kurosh-Amitsur radical in the variety of Γ-near-rings.     

Universal norms in Γ-extensions

A simple new proof is given for a theorem of L. V. Kuz'min (Ref. Zh. Mat., 1972, 10A239) on the structure of the group of universal norms in -extensions, using his theory of Tate modules.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 251–255, 1977.     

Hereditary Critical Ω-Compositional Formations

We present a solution of the Shemetkov problem (on the investigation of critical formations) for piecewise-compositional hereditary formations.     

Some Results Concerning Hereditary Algebras

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Outer Γ-Convexity in Vector Spaces

A subset S of some vector space X is said to be outer Γ-convex w.r.t. some given balanced subset Γ ? X if for all x ₀, x ₁ ? S there exists a closed subset Λ ? [0,1] such that {x _λ | λ ? Λ} ? S and [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ, where x _λ: = (1 ? λ)x ₀ + λ x ₁. A real-valued function f:D → ? defined on some convex D ? X is called outer Γ-convex if for all x ₀, x ₁ ? D there exists a closed subset Λ ? [0,1] such that [x ₀, x ₁] ? {x _λ | λ ? Λ} + 0.5 Γ and f(x _λ) ≤ (1 ? λ)f(x ₀) + λ f(x ₁) holds for all λ ? Λ. Outer Γ-convex functions possess some similar optimization properties as these of convex functions, e.g., lower level sets of outer Γ-convex functions are outer Γ-convex and Γ-local minimizers are global minimizers. Some properties of outer Γ-convex sets and functions are presented, among others a simplex property of outer Γ-convex sets, which is applied for establishing a separation theorem and for proving the existence of modified subgradients of outer Γ-convex functions.     

Notes on Brown-McCoy-radicals for Γ-near-rings

The Brown-McCoy radical is known to be an ideal-hereditary Kurosh-Amitsur radical in the variety of zerosymmetric near-rings. We define the Brown-McCoy and simplical radicals, and , respectively, for zerosymmetric -near-rings. Both and are ideal-hereditary Kurosh-Amitsur radicals in that variety. IfM is a zerosymmetric -near-ring with left operator near-ringL, it is shown that , with equality ifM has a strong left unity. is extended to the variety of arbitrary near-rings, and and are extended to the variety of arbitrary -near-rings, in a way that they remain Kurosh-Amitsur radicals. IfN is a near-ring andA N, then , with equality ifA if left invariant.     

The Γ-algorithm and some applications

In this paper the power of the Γ-algorithm for obtaining the dual of a given cone and some of its multiple applications is discussed. The meaning of each sequential tableau appearing during the process is interpreted. It is shown that each tableau contains the generators of the dual cone of a given cone and that the algorithm updates the dual cone when new generators are incorporated. This algorithm, which is based on the duality concept, allows one to solve many problems in linear algebra, such as determining whether or not a vector belongs to a cone, obtaining the minimal representations of a cone in terms of a linear space and an acute cone, obtaining the intersection of two cones, discussing the compatibility of linear systems of inequalities, solving systems of linear inequalities, etc. The applications are illustrated with examples.     

Fuzzy generalized Bi-ideals of Γ-semigroups

In this paper, we introduce the concept of fuzzy generalized bi-ideal of a ??-semigroup, which is an extension of the concept of a fuzzy bi-ideal of a ??-semigroup and characterize regular ??-semigroups in terms of fuzzy generalized biideals.     

On “P” property and the column-W property

P-matrices play an important role in the well-posedness of a linear complementarity problem (LCP). Similarly, the well-posedness of a horizontal linear complementarity problem (HLCP) is closely related to the column-W property of a matrix k-tuple.In this paper we first consider the problem of generating P-matrices from a given pair of matrices. Given a matrix pair (D, F) where D is a square matrix of order m and matrix F has m rows, “what are the conditions under which there exists a matrix G such that (D + FG) is a P-matrix?”. We obtain necessary and sufficient conditions for the special case when the column rank of F is m ? 1. A decision algorithm of complexity O(m²) to check whether the given pair of matrices (D, F) is P-matrisable is obtained. We also obtain a necessary and an independent sufficient condition for the general case when rank(F) is less than m ? 1.We then generalise the P-matrix generating problem to the generation of matrix k-tuples satisfying the column-W property from a given matrix (k + 1)-tuple. That is, given a matrix (k + 1)-tuple (D₁, … ,D_k, F), where D_js are square matrices of order m and F is a matrix having m rows, we determine the conditions under which the matrix k-tuple (D₁ + FG₁, … ,D_k + FG_k) satisfies the column-W property. As in the case of P-matrices we obtain necessary and sufficient conditions for the case when rank(F) = m ? 1. Using these conditions a decision algorithm of complexity O(km²) to check whether the given matrix (k + 1)-tuple is column-W matrisable is obtained. Then for the case when rank(F) is less than m ? 1, we obtain a necessary and an independent sufficient condition.For a special sub-class of P-matrices we give a polynomial time decision algorithm for P-matrisability. Finally, we obtain a geometric characterisation of column-W property by generalising the well known separation theorem for P-matrices.