Note on Independence Functions
Note on Independence Functions
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DOI：
10.1112/plms/s3-7.1.300
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发表时间：
1957
影响因子：
1.8
通讯作者：
R. Rado
中科院分区：
文献类型：
--
作者：
R. Rado
几位作者已经研究了代数中独立性概念的公理结构。特别是哈斯勒·惠特尼 (Hassler Whitney) (1) 对这一主题进行了深入的研究。进一步的参考文献在（2）中给出。在本注释中，在给定的集合8上，将通过下面(3)中给出的条件(1)-(5)来定义独立函数(f-函数)。如果 是 8 上的 /-函数，并且 x0,..., xn_1 是 8 的元素，那么该关系意味着这些元素是“独立的”，并且 /= 0 表示它们是“相关的”。在定理 1 中，将建立 /-函数的新特征，该特征很有趣，因为它不包含对所涉及集合的基数的引用。 Whitney (1) 提出了基本表示问题：给定有限集 8（惠特尼术语中的拟阵）上的 /-函数 /，满足什么条件才能存在 S 的 x-> v (x) 映射到某个域 K 上某个固定维度的所有向量的集合，使得 f {xQ,..., xn_^)= 0 当且仅当向量 v (x0),..., v {xn_^) 是线性的依赖于 K1 惠特尼解决了当 K 是 2 个元素的伽罗瓦域 GF (2) 时的问题。在本文中，将给出惠特尼结果的简单证明（定理 2）。表示问题不会被解决，但下面的结果将被证明，这可能会对这个问题有所启发。（i）每个在域 K 上可表示的函数也可以在 K 中包含的素数域的某些有限代数扩展上表示，并且对于无限多个素数 p，也可以在域 GF (p) 上表示（定理 4）。
THE axiomatic structure of the notion of independence in algebra has been investigated by several authors. In particular, Hassler Whitney (1) has made a thorough study of this topic. Further references are given in (2). In the present note independence functions (/-functions) will be defined, on a given set 8, by the conditions (l)-(5) below given in (3). If/is an/-function on 8 and x0,..., xn_1 are elements of 8 then the relation means that these elements are'independent', and/= 0 that they are'dependent'.In Theorem 1 a new characterization of/-functions will be established which is of interest in that it contains no reference to the cardinal numbers of the sets involved. Whitney (1) has formulated the fundamental representation problem: given an/-function/on a finite set 8 (a matroid, in Whitney's terminology), what are the conditions in order that there should exist a mapping x-> v (x) of S into the set of all vectors of some fixed dimension over some field K such that f {xQ,..., xn_^)= 0 if and only if the vectors v (x0),..., v {xn_^) are linearly dependent over K1 Whitney solved the problem in the case when K is the Galois fieldGF (2) of 2 elements. In this note a simple proof of Whitney's result will be given (Theorem 2). The representation problem will not be solved but the following results will be proved which may shed some light on the problem.(i) Every/-function which is representable over a field K is also representable over some finite algebraic extension of the prime-field contained in K, and also over the field GF (p), for infinitely many primes p (Theorem 4).