Notes on (m,n) bi-Γ-ideals in Γ-semigroups

The notions of quasi-ideals of rings and semigroups were introduced by Steinfeld (Acta Math. Acad. Sci. Hung. 4:289–298, 1953 and Publ. Math. (Debr.) 4:262–275, 1956) respectively. The notion of Γ-semigroups was introduced by Sen (Proceeding of International Symposium on Algebra and Its Applications, Decker Publication, New York, 1981). Further the notion of (m,n) ideals of semigroups was introduced by Lajos (Acta Sci. Math. 22:217–222, 1961). Later on (m,n) quasi-ideals and (m,n) bi-ideals were widely studied in various algebraic structures viz. semigroups, rings and near-rings etc. In this paper we have defined (m,n) quasi-Γ-ideal and (m,n) bi-Γ-ideal in Γ-semigroup. Including other results, we have shown that if Q is a minimal (m,n) quasi-Γ-ideal in Γ-semigroup S then intersection of minimal m-left Γ-ideals and minimal n-right Γ-ideals is again a minimal (m,n) quasi-Γ-ideals.     

Ideals in (m + 1)-semigroups

Summary The following memoir is concerned with various idealtheoretic aspects of the theory of polyadic semigroups. Many of the results are generalizations of known theorems in the theory of ordinary or 2-semigroups.     

Projection representable relations on Menger (2, n)-semigroups

Abstract characterizations of relations of nonempty intersection, inclusion end equality of domains for partial n-place functions are presented. Representations of Menger (2, n)-semigroups by partial n-place functions closed with respect to these relations are investigated.     

On the functional equationS(m,n)F[n/(m,n)] =F(n)h[n/(m,n)]

In this paper, we discuss the pairs (f, h) of arithmetical functions satisfying the functional equation in the title, whereF is the product off andh under the Dirichlet convolution; that is,F(n) = Σ _d|n ?(d)h(n/d) andS(m n) = Σd|(m, n) ?(d)h(n/d). The well-known Hölder's identity is a special case of this functional equation (?(n) =n, h(n) = μ(n)). We also generalize the functional equation in the title to any arbitrary regular arithmetical convolution and discuss the pairs of solutions (f, h) of the generalized functional equation and pose some problems relating to the characterization of all pairs of solutions.     

(m,n)模糊超环

介绍(m,n)超环等一些相关概念,之后将(m,n)超环模糊化,给出(m,n)模糊超环的定义,初步探讨(m,n)模糊超环的结构和性质,分析(m,n)模糊超环在同态下的不变性。     

广义(m,n)超环

本文研究了广义(m,n)超环,n元正则关系以及n元强正则关系等的一些性质.利用广义(m,n)超环间的同态关系以及正则和强正则关系,得到了(m,n)子超环和(m,n)超理想的不变性,广义(m,n)超环的商结构,以及构成商超环和商环的充分必要条件,推广了文献[5]的一些结果.     

Intrinsic normalizations of seminonholonomic complexes SNGr(m,n, (m+1) (n−m)−ϱ)

V. Kapsukas Vilnius State University. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 28, No. 2, pp. 299–314, April–June, 1988.     

Equationally complete (m,n) rings

It is shown that every super-simple (m, n) ring is equationally complete. The atomic varieties of (m, 2) rings and the atomic varieties of (2,n) rings are completely determined.     

Characterizations of ( m,n )-Jordan Derivations and ( m,n )-Jordan Derivable Mappings on Some Algebras

Let R be a ring, M be a R-bimodule and m, n be two fixed nonnegative integers with m + n = 0. An additive mapping δ from R into M is called an(m, n)-Jordan derivation if(m +n)δ(A~2) = 2 mAδ(A) + 2nδ(A)A for every A in R. In this paper, we prove that every(m, n)-Jordan derivation with m = n from a C*-algebra into its Banach bimodule is zero. An additive mappingδ from R into M is called a(m, n)-Jordan derivable mapping at W in R if(m + n)δ(AB + BA) =2mδ(A)B + 2 mδ(B)A + 2 nAδ(B) + 2 nBδ(A) for each A and B in R with AB = BA = W. We prove that if M is a unital A-bimodule with a left(right) separating set generated algebraically by all idempotents in A, then every(m, n)-Jordan derivable mapping at zero from A into M is identical with zero. We also show that if A and B are two unital algebras, M is a faithful unital(A, B)-bimodule and U = [A M N B] is a generalized matrix algebra, then every(m, n)-Jordan derivable mapping at zero from U into itself is equal to zero.     

On (n,m)-Convex Sets

We investigate the class of generalized convex sets on Grassmann manifolds, which includes known generalizations of convex sets for Euclidean spaces. We extend duality theorems (of polarity type) to a broad class of subsets of the Euclidean space. We establish that the invariance of a mapping on generalized convex sets is equivalent to its affinity.