针对膨胀土胀缩等级评定中单指标和多指标评价方法的不足,将液限、总胀缩率、塑性指数、天然含水率以及自然膨胀率作为膨胀土胀缩等级评价的指标,采用熵值法评价各不同指标的权重,以理想点原理为依托,提出了膨胀土胀缩等级评价的熵值-理想点模型。首先建立胀缩等级的理想点和反理想点矩阵,通过构造距离函数来表征样本与不同胀缩等级的理想点和反理想点距离,然后引入理想点贴近率的概念评价样本等级,即理想点距离越小,贴近率越小,样本越接近于该等级。该模型在某工程膨胀土胀缩等级评价中的应用表明:采用熵值-理想点模型评价的膨胀土的胀缩等级与实际情况吻合较好,说明该模型具有合理性和实用性。

Due to the shortcomings of single index and multi-index evaluation method in grade of shrink and expansion for the expansive soil,the liquid limit and swell-shrink total ratio,plasticity index,natural moisture content and natural expansion rate were selected as the evaluation index of the grade of shrink and expansion for the expansive soil evaluation,it was evaluated the weights of different indexes by using entropy method, based on the principle of ideal point,the model of entropy-ideal point method of shrink and expansion for the expansive soil had been established. It was firstly created ideal point and negative ideal point matrix about the grade of shrink and expansion,by establishing a distance function to char-acterize of samples with different grade of shrink and expansion ideal point and the negative ideal point distance,and then introduced the concept of ideal point close to the rate of evaluation sample grades,i. e. the ideal point distance was smaller,close to the rate is s

在N (2、2、0)代数中引入了素理想的概念；利用模糊集理论，提出了点态化(?、?úq(λ、μ))-模糊素理想和广义模糊素理想的概念，讨论了两者之间的关系；得到了(?、?úq(λ、μ))-模糊素理想的一些等价刻画及其相关性质；研究了(?、?úq(λ、μ))-模糊素理想同态像与同态原像的基本性质。

The concept of prime ideals of the N (2,2,0) algebra is introduced;with the help of fuzzy sets, the concepts of point-wise (∈,∈ Vq(λ,μ))-fuzzy prime ideals and generalized fuzzy prime ideals are presented, and the relationship between them is discussed;some equivalent descriptions and properties of the (∈,∈ Vq(λ,μ))-fuzzy prime ideals are obtained;some basic proper-ties of homomorphic image and homomorphic preimage of the(∈,∈ Vq(λ,μ))-fuzzy prime ideals are studied.

引入N(2,2,0)代数结合理想的概念。利用(∈,∈∨q(λ,μ))-模糊代数,给出了N(2,2,0)代数广义(λ,μ)-模糊结合理想和点态化模糊结合理想的概念,讨论了两者之间的等价关系;给出了(∈,∈∨q(λ,μ))-模糊结合理想的一些等价刻画,研究了其相关性质;讨论了(∈,∈∨q(λ,μ))-模糊结合理想的同态像与同态原像的关系;还研究了(∈,∈∨q(λ,μ))-模糊结合理想的直积。

The concept of associative ideal of the N (2,2,0) algebra is introduced;with the help of (∈,∈∨q(λ,μ))-fuzzy algebra, the notions of generalized (λ,μ)-fuzzy associative ideal and pointwise fuzzy associative ideal are proposed, and the relationship between them is also discussed;besides, this paper presents some equivalent descriptions of the (∈,∈∨q(λ,μ))-fuzzy associative ideal, and discusses the related properties;some basic properties of homomorphic image and homomor-phic preimage of the (∈,∈∨q(λ,μ))-fuzzy associative ideal are discussed. Direct product of the (∈,∈∨q(λ,μ))-fuzzy asso-ciative ideal is also studied.

针对属性值为犹豫模糊元的决策问题,提出一种基于粗糙集理论的多属性决策方法.首先,依据属性值与理想点的贴近度和给定的阀值得到判断矩阵;然后,根据判断矩阵对属性集进行约简,确定属性权重;最后,基于TOPSIS思想,计算各方案与理想点的综合贴近度,得到方案的优劣次序,并通过算例分析表明了该方法的有效性.

A decision method based on rough sets is proposed for multi-attribute decision making, in which the attribute values are in the form of hesitant fuzzy elements. Firstly, the discernibility matrix is acquired according to the given threshold and the similarity degree between attribute values and ideal points. Then, the attribute reduction is operated and the weight of core attribute is determined by the discernibility matrix. Finally, the weighted similarity degree of every alternative with ideal points is displayed to rank all the alternatives. An example is given to illustrate the effectiveness of the proposed method.

设 G 是有限群,R（G）为 G 的复表示环,I（G）为其增广理想。对第一类点群（特殊正交群 SO3（R）的有限子群）和任意的自然数 n,给出了增广理想的 n 次幂 I n（G）作为自由交换群的基底,并确定了其增广商群In（G）/ I n＋1（G）的结构。

Let G be a finite group, R(G) its complex representation ring,and I(G) its augmentation ideal. This paper obtained an explicit Z-basis of I n(G) and determined the isomorphism class of the n-th augmentation quotient I n(G) / I n+1(G) for the first type of point group (i. e. finite subgroup of the special orthogonal group SO3(R)) and for each positive integer n.