A note on lattice ordered ‐algebras and Perron–Frobenius theory

A classical result of Sherman says that if the space of self‐adjoint elements in a ‐algebra is a lattice with respect to its canonical order, then is commutative. We give a new proof of this theorem which shows that it is intrinsically connected with the spectral theory of positive operator semigroups. Our methods also show that some important Perron–Frobenius like spectral results fail to hold in any non‐commutative ‐algebra.     

对交换BCI-代数的几点注记

得到交换 Ｂ Ｃ Ｉ代数 Ｘ 的一些进一步性质．主要结果有：（ｉ）　给出 Ｘ 的交换性的两个特征;（ｉｉ）　证明对任意 ｃ∈ Ｘ,ｃ 的初始段 Ａ（ｃ）关于 Ｂ Ｃ Ｉ序≤构成一个分配格（ Ａ（ｃ）;≤）;（ｉｉｉ）　指出 Ｘ 未必是拟交换的     

Higher homotopy commutativity of -spaces and the permuto-associahedra

In this paper, we give a combinatorial definition of a higher homotopy commutativity of the multiplication for an -space. To give the definition, we use polyhedra called the permuto-associahedra which are constructed by Kapranov. We also show that if a connected -space has the finitely generated mod cohomology for a prime and the multiplication of it is homotopy commutative of the -th order, then it has the mod homotopy type of a finite product of Eilenberg-Mac Lane spaces s, s and s for .

     

Commutativity of Rings with Constraints Involving a Subset

Suppose that R is an associative ring with identity 1, J(R) the Jacobson radical of R, and N(R) the set of nilpotent elements of R. Let m 1 be a fixed positive integer and R an m-torsion-free ring with identity 1. The main result of the present paper asserts that R is commutative if R satisfies both the conditions(i) [x ^m, y ^m] = 0 for all and(ii) [(xy) ^m + y ^m x ^m, x] = 0 = [(yx) ^m + x ^m y ^m, x], for all This result is also valid if (i) and (ii) are replaced by (i) [x ^m, y ^m] = 0 for all and (ii) [(xy) ^m + y ^m x ^m, x] = 0 = [(yx) ^m + x ^m y ^m, x] for all Other similar commutativity theorems are also discussed.     

Lagrange equation of a class of nonholonomic systems

Making use of conclusions from[1]:(1)d-δoperations are commutative;(2)theAppell-Chetaev condition restricting virtual displacements is superfluous,the present paperderives the Lagrange equation without multipliers for a class of first-order nonlinearnonholonomic dynamical systems by means of variational principle.This kind of equationsis new.     

几类亚交换序半群

本文研究几类亚交换序半群的性质,并将具有单位元的亚交换序半群的一些结果扩张到这几类亚交换序半群上,使得这些结果更加细化,其中主要证明了以下定理：伪交换序半群可以分解成阿基米德序半群的半格．并且,一般来说,这种分解不是唯一的．     

Probabilistic aspects of ZM-groups

In this paper we study probabilistic aspects such as the (cyclic) subgroup commutativity degree and different types of factorization numbers of ZM-groups. We show that these quantities can be computed using the sizes of the conjugacy classes of these groups. In the end, we point out a class of groups whose (cyclic) subgroup commutativity degree vanishes asymptotically.     

Universal homogeneous derivations of graded ϵ-commutative algebras

Let K be a commutative ring, let ? be an abelian group, and let ?:?x?→K be a commutation factor over ?.A ? graded K-algebra is said to be ?-commutative if its ?-bracket is identically zero, (K,?) derivations from a given ?-commutative ?-graded K-algebra A into bimodules are studied. It is proved that for each λ?? there exists a universal initial (k,?)-derivation of degree λ of A. For each λ?? a natural module of (K, ?, λ)-differentials of A along with a differential map is constructed. It is proved that each derivation of A canonically equipps this module with a structure of differential module. Applications and examples are given. It is shown that the first order exterior differentials which are known from the theory of smooth graded manifolds are universal initial homogeneous derivations of the sort considered hereby.     

Nonlinear Strong Commutativity Preserving Maps on Prime Rings

Let 𝒜 be a unital prime ring containing a nontrivial idempotent P. Assume that Φ: 𝒜 → 𝒜 is a nonlinear surjective map. It is shown that Φ preserves strong commutativity if and only if Φ has the form Φ(A) = αA + f(A) for all A ∈ 𝒜, where α ∈ {1, ?1} and f is a map from 𝒜 into 𝒵(𝒜). As an application, a characterization of nonlinear surjective strong commutativity preserving maps on factor von Neumann algebras is obtained.