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  quasi-domain
    The fuzzy matrices defined on fuzzy scalar quasi-domain are also studied in the chapter. It is proved that fuzzy linear transformations can be characterized by fuzzy matrices mentioned above.
    提出了模糊数量拟域上的矩阵——模糊矩阵的概念,并讨论了模糊矩阵与模糊线性变换之间的关系.
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If ■ is the set of left multiplications (SLM) of a (left) quasi-field Q, then H■(H~(-1)■),■~(-1) and A-■(A■,A-I■)are also SLMs of quasi-fields. They are called the three fundamental derivatizations of ■.When the SLM of a quasi-field Q′ is obtained by a series of fundamental derivatizations of the SLM of Q,Q′ is said to be a derivatization of Q.A derivatization of Q is equipollent with Q,i.e.,they define isomorphic projective planes. If a quasi-field Q is obtained by changing the left multiplication on a subset...

If ■ is the set of left multiplications (SLM) of a (left) quasi-field Q, then H■(H~(-1)■),■~(-1) and A-■(A■,A-I■)are also SLMs of quasi-fields. They are called the three fundamental derivatizations of ■.When the SLM of a quasi-field Q′ is obtained by a series of fundamental derivatizations of the SLM of Q,Q′ is said to be a derivatization of Q.A derivatization of Q is equipollent with Q,i.e.,they define isomorphic projective planes. If a quasi-field Q is obtained by changing the left multiplication on a subset ■ of a field ■,Q is called a distortion of ■(or Q is a distorted field) and ■ is called the scion of Q.André systems (and generalized André systems) are examples of distorted fields.Let Q_1 and Q_2 be two distortions of a finite field ■ with scions ■_1 and ■_2 respectively.If ■_1∩■=φ,a distorted field Q, called the grafting of Q_1 and Q_2,can be constructed with ■_1∪■_2 as its scion. Under additional conditions,grafting of distortions of an infinite field is also possible. As examples,varied derivatizations of André systems are obtained, among them are the Hall quasi-fields (finite or infinite).Thus,the fact that Hall planes are isomorphic to certain André planes is obvious. By grafting varied derivatizations of varied André systems,several new classes of quasi-fields are constructed explicitly.Especially,a large class of generalized André systems (which are finite by definition)is obtained as the grafting of André systems and thus can be generalized to infinite cases.

设■是拟域Q的左乘集,则(i)H■(H~(-1)∈■),(ii)■~(-1)及(iii)A-■(A∈■,A-I∈■)仍是拟域的左乘集,上面三种变换称为基本衍生。如Q′的左乘集可由Q的左乘集经一串基本衍生而得到,则称 Q′是Q的衍生。Q的衍生与Q等效,即它们定义的射影平面是同构的。我们对 André拟域作各种衍生,其中特别我们得到了 Hall 拟域(有限或无限)。一个域■如果在它的一部分■上改变乘法成为一拟域 Q,则称Q是■的畸变,或简称 Q 为一畸变域,■称为 Q 的穗部,有限域■的二个畸变Q_1及Q_2,它们的穗部分别是■及■,如果■∩■=φ,则可作畸变域Q 以■∪■为穗部,称为Q_1与Q_2的嫁接,记为 Q=Q_1∨Q_2(对无限情形,如果增加适当条件,也可作嫁接)。用不同的 André拟域的不同的衍生作嫁接,我们得到了多类新的拟域。特别,广义 André拟域中很大一类可用 André拟域嫁接得到,从而可推广到无限情形。

This paper furnishes an answer to the problem raised from[1] “Are the seven planes ccordinatedby non-Desarguesian Andre quasifield all possibly isOmorphic?”It is proved that there are onlythree non-Desarguesian Andre planes of order 25。

本文回答P.R.Hugbo提出的一个问题:“是否由7个非berguereanAndre拟域坐标的平面可能全部同构?”给出的结论是:有且只有3个25阶非DesarguesianAndre平面。

In this paper, we considered the matrix equation AXB+CYD=0 in an arbitrary skew field, and presented the expressions of solutions of the equation and a necessary and sufficient condition for the existence of a unique solution of the equation.

给出了拟域上矩阵方程 AXB+CYD=0的通解表达式及其仅有零解的一个充要条件

 
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