经典逻辑度量空间中的边角关系  

Relationship of lenghts and angles in the classical logic metric space

作　　者：胡明娣[1] 楼志刚[2] 

机构地区：[1]陕西师范大学数学研究所,陕西西安710062 [2]西安理工大学机械与精密仪器工程学院,陕西西安725000

出　　处：《西北大学学报（自然科学版）》2011年第2期205-209,共5页Journal of Northwest University（Natural Science Edition）

基　　金：国家自然科学基金资助项目(10771129);陕西师范大学研究生培养创新基金资助项目(2009CXB006)

摘　　要：目的研究逻辑度量空间的内蕴结构,讨论其中三角形的结构及相关性质。方法利用计量逻辑学理论中建立的距离函数进行计算。结果首先证明了在经典逻辑度量空间([F(S)],ρ)中存在等边多边形,直角三角形等特殊图形。其次证明了不存在边长大于或等于2/3的等边三角形,但存在边长可任意接近2/3的等边三角形。同时证明了Lindenbaum代数上的反射变换φ*和平移变换ηG保持等边三角形、直角三角形的边角关系不变。最后证明了在经典逻辑度量空间中三逻辑公式构成的三角形中,内角的余弦在[0,1]中稠密,即,它们的内角在[0,π2]上稠密分布。结论等边三角形的边长可任意接近2/3,但是逻辑度量空间中不存在边长大于或等于2/3的等边三角形。并且,Lindenbaum代数上的反射变换和平移变换保持等边三角形、直角三角形的边角关系不变。以上结论为进一步讨论和找寻经典逻辑度量空间中的基本结构奠定了基础。Aim To investigate the intrinsic structure of the classical logic metric space,discuss the structure of equilateral triangles and related properties.Methods The metric function proposed in quantitative logic was used as basic tool to develop computations.Results It is proved that there are some special graphs like Equilateral polygons and right triangles in the classical logic metric space.It is proved that there does not exist any equilateral triangle with length of 2/3 or more than 2/3,but there exist abundance of equilateral triangles with length arbitrarily close to 2/3.There exists an isometric reflexion transform and parallel trnsform which preserve the character of the equilateral triangle unchanged on the Lindenbaum algebras.Lastly,it is proved that,in the classical logic metric space,the values of cosine of an inside angle of a triangle constituted by three logic formulaes is dense in the unit interval ,i.e.,the degree of the inside angles of triangles is dense in .Conclusion In the classical logic metric space,there exist abundance of equilateral triangles with length arbitrarily close to 2/3 but there cloesn′t exist any equilateral triangle with length of 2/3 or more than 2/3,and,both of the reflexion transform and the parallel transform can preserve the character of the equilateral triangle and the right triangle unchanged.The above conclusions lay the foundation for the studying of the basic structure of the classical logical metric space.

关 键 词：经典逻辑度量空间 等边三角形 LINDENBAUM代数 等距变换 内角 

分 类 号：O159[理学—数学]