The bracket ring of a combinatorial geometry. I
The bracket ring of a combinatorial geometry. I
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组合几何的支架环。
DOI：
10.1090/s0002-9947-1975-0387095-9
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发表时间：
1975
影响因子：
1.3
通讯作者：
N. White
中科院分区：
文献类型：
--
作者：
N. White
括号环是在任意组合几何G上构造的广义行列式的环，称为括号。括号满足行列式的几个熟悉的性质，包括合冲，这相当于拉普拉斯的扩展由未成年人。证明了括号环是G在两种意义下的泛配位对象。首先，G的协调对应于环到域的同态，从而将G的协调的研究归结为确定括号环的素理想结构。其次，G在它自己的括号环上有一个类似于坐标化的表示，这使得线性代数的一些熟悉的结果，包括Cramer规则，得到了有趣的推广。另一种形式的syzygies然后推导和应用的问题找到一个标准形式的任何元素的括号环。最后，我们证明了几何之间的几个重要关系，即正交性，子几何和收缩，直接反映在括号环的结构。
The bracket ring is a ring of generalized determinants, called brackets, constructed on an arbitrary combinatorial geometry G. The brackets satisfy several familiar properties of determinants, including the syzygies, which are equivalent to Laplace's expansion by minors. We prove that the bracket ring is a universal coordinatization object for G in two senses. First, coordinatizations of G correspond to homomorphisms of the ring into fields, thus reducing the study of coordinatizations of G to the determination of the prime ideal structure of the bracket ring. Second, G has a coordinatization-like representation over its own bracket ring, which allows an interesting generalization of some familiar results of linear algebra, including Cramer's rule. An alternative form of the syzygies is then derived and applied to the problem of finding a standard form for any element of the bracket ring. Finally, we prove that several important relations between geometries, namely orthogonality, subgeometry, and contraction, are directly reflected in the structure of the bracket ring.