Absolute Valued Algebras Satisfying ( x 2, x 2, x 2) = 0

Abstract

Let A be an absolute valued algebra. We prove that if A satisfies the identity (x ², x ², x ²) = 0 for all x in A, and contains a central idempotent e, that is ex = xe for all x in A, then A is finite dimensional. This result enables us to prove that if A satisfies (x ², x ², x ²) = 0 and admits an involution then A is finite dimensional. To show that our assumptions on A are essential we recall that in El-Mallah [El-Mallah, M. L. (1988). Absolute valued algebras with an involution. Arch. Math. 51:39–49] it was shown that the existence of a central idempotent in A is not a sufficient condition for A to be finite dimensional; and the example given in El-Mallah [El-Mallah, M. L. (2003). Semi-algebraic absolute valued algebras with an involution. Comm. Algebra 31(7):3135–3141] shows that there exist infinite dimensional semi-algebraic absolute valued algebras satisfying the identity (x ², x ², x ²) = 0.     

Semi-algebraic Absolute Valued Algebras with an Involution

Abstract

Let A be an absolute valued algebra. In El-Mallah (El-Mallah,M. L. (1988). Absolute valued algebras with an involution. Arch. Math. 51: 39–49) we proved that,if A is algebraic with an involution,then A is finite dimensional. This result had been generalized in El-Amin et al. (El-Amin,K.,Ramirez,M. I.,Rodriguez,A. (1997). Absolute valued algebraic algebras are finite dimensional. J. Algebra 195:295–307),by showing that the condition “algebraic” is sufficient for A to be finite dimensional. In the present paper we give a generalization of the concept “algebraic”,which will be called “semi-algebraic”,and prove that if A is semi-algebraic with an involution then A is finite dimensional. We give an example of an absolute valued algebra which is semi-algebraic and infinite dimensional. This example shows that the assumption “with an involution” cannot be removed in our result.     

Remarks on the lattices of fuzzy subsets of a groupoid

In this paper we give a necessary and sufficient condition for a groupoid D such that the sup-min product is distributive over arbitrary intersection of fuzzy subsets of D, and correct some results from the paper [S. Ray, The lattice of all idempotent fuzzy subsets of a groupoid, Fuzzy Sets and Systems 96 (1998) 239–245]. Also, we prove that the set of all idempotent fuzzy sets forms a complete lattice, which is a complete join-sublattice of the lattice of all fuzzy subgroupoids. This result extends the corresponding result from the above mentioned paper.     

MgCl2/R'nAl(OR)3-n: an excellent activator/support for transition-metal complexes for olefin polymerization

A new and effective method for the activation, and simultaneously, immobilization of bis(phenoxyimine) early-transition-metal complexes for olefin polymerization (known as FI catalysts), which makes use of MgCl(2)/R'(n)Al(OR)(3-n) as an activator/support, has been developed. Ti-, Zr-, and V-FI catalysts combined with this MgCl(2)-based compound can form highly active MgCl(2)-supported single-site catalysts capable of demonstrating superior catalytic properties, compared to the corresponding homogeneous methylaluminoxane- (Ti- and Zr-FI catalysts) or alkylaluminum-activation systems (V-FI catalysts), in terms of their catalytic activity, molecular weight, stereoselectivity, and comonomer incorporation. Additionally, these new catalysts can produce polymers of significant morphology with high efficiency. Notably, the MgCl(2)-based compounds can also effectively activate and immobilize the early-to-late transition-metal complexes that have emerged recently. Thus, the application of MgCl(2)-based compounds as activators/supports for transition-metal complexes for olefin polymerization provides a conceptually new strategy for the development of methylaluminoxane- and borate-free, high-performance, single-site catalysts capable of controlling polymer morphology.     

平衡半拟补Ockham代数的滤子

一个平衡半拟补Ockham代数(简称bdpO-代数)是一个〈2,2,1,1,0,0〉类型的代数(L;∧,∨,f,*,0,1).其中(L;f)是Ockham代数,(L;*)是半拟补代数,且一元运算f和*由恒等式[f(x)]*=f2(x)与f(x*)=x**所连结.本文讨论bdpO-代数中滤子的性质及其同余特征,并刻画某些同余一致与同余凝聚性质.     

The derived category of an algebra with radical square zero

Koszul duality and covering theory are combined to realize the bounded derived category 𝒟 of an algebra with radical square zero as a certain orbit category of the bounded derived category of finitely presented representations of an associated infinite quiver. As a consequence, the possible shapes of the connected components of the Auslander–Reiten quiver of 𝒟 are described.     

On Eilenberg–Moore algebras induced by chains

Let S be a fixed topological space. The contravariant Hom functor given by C(X) = has an adjoint specified, on sets, by P(A)= S ^A and the composite, , is a Monad on the category of sets. In this paper we characterize the category of Eilenberg–Moore Algebras associated with M in the special case where S is a linearly ordered space in its specialization order. The characterization is presented in terms of the notion of a dual frame which admits a C(S)-action. Received June 11, 1998, accepted in final form March 2, 1999     

On-line galvanic cell generated electrochemiluminescence determination of acyclovir based on the flow injection sampling

A new simple, rapid and cost effective flow injection (FI) electrochemiluminescence (ECL) method was described for the determination of acyclovir (9,2-hydroxyethoxy) methyl guanine. In the presence of acyclovir, the luminol electrochemiluminescence generated by a mini on-line galvanic cell could be greatly sensitized in alkaline medium. The relative electrochemiluminescence intensity was found to increase linearly with increasing concentration of acyclovir, which was corroborated by the calculated correlation coefficient value of 0.9994 (n=7). The limit of detection was 1.6×10⁻⁷ mol l⁻¹ and the limit of quantification was 7.9×10⁻⁷ mol l⁻¹. The proposed method was applied to the determination of acyclovir in pharmaceutical formulations. The reliability of the assay method was established by parallel determination and by standard-addition method. Experiment results demonstrated the described mini analysis system, while being simple and less time consuming, was accurate, precise and reproducible (R.S.D. = 1.6%, recoveries = 99 - 103%). Further experiments indicated that there was no significant difference between the results obtained by the proposed and official methods.     

烯烃聚合高性能FI催化剂的进展

FI催化剂系一近几年发展起来的烯烃聚合高性能催化剂。以它的极高的活性、聚合物分子结构的可控性、多重活性聚合等特性而引人注目。本评述首先进行该催化剂的化学描述,然后分别介绍该催化剂的主要特征、活性中心结构、聚合机理和应用,重点介绍该催化剂的五大特征。