On weakly ordered systems
On weakly ordered systems
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在弱有序系统上
DOI：
10.1090/s0002-9904-1946-08518-3
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发表时间：
1946
影响因子：
1.3
通讯作者：
M. Richardson
中科院分区：
文献类型：
--
作者：
M. Richardson
陈述“x>y”可以读作“x支配y”。传递性不是假设的;传递弱序系统是偏序系统。弱序系统的解是指D的一个元素集V，使得（a）X ∈ LV和y ∈ V意味着x<y是假的，（B）#∈ D-F意味着对某些y ∈ V有3>>#。Morgenstern，Theory of games and economic behavior，Princeton，1944，其中证明了严格非循环的弱序系统具有唯一的解，并给出了其构造。这一结果表明，一般弱序系统的解的存在性和唯一性的条件的问题。最简单的例子表明，如果存在圈，则在所有情况下都不能期望解的存在性和唯一性。例如，三个元素的系统a>B>c>a没有解，而四个元素的系统a>B>c>d>a有两个解（a，c）和（bf d）。本文的目的是证明某些非循环系统解的存在性。证明本身将为解决方案提供一种构造方法。策梅洛的选择公理，良序定理，超限归纳将被使用。下面给出的结果是对上述一般问题的贡献，而不是对博弈论的贡献。因为下面定理的假设完全排除了传递性;也就是说，它排除了三个元素a、B、c的存在，使得a>B，B>c，并且a>c。这个限制对于博弈论来说太严格了，就像传递性的假设一样。这个问题对于弱序系统仍然是开放的，这些系统不是严格无环的，但也不满足下面定理的假设。
The statement "x>y" may be read "x dominates y." Transitivity is not assumed ; a transitive weakly ordered system is a partially ordered system. By a solution of a weakly ordered system is meant a set V of elements of D such that (a) XÇLV and y £ V implies x<y is false and (b) #£D— F implies 3>># for some yÇ~ V. The concept of solution was introduced in J. von Neumann and O. Morgenstern, Theory of games and economic behavior, Princeton, 1944, where it is proved that a weakly ordered system which is strictly acyclic possesses a solution which is unique, and for which a construction is given. This result suggests the problem of finding conditions for the existence and uniqueness of solutions of weakly ordered systems in general. The simplest examples show that if cycles exist neither the existence nor the uniqueness of solutions can be expected in all cases. For example, the system of three elements a>b>c>a has no solution, while the system of four elements a>b>c>d>a has the two solutions (a, c) and (bf d). The purpose of this note is to prove the existence of solutions for certain non-acyclic systems. The proof will itself provide a method of construction for the solutions. Zermelo's axiom of choice, the well-ordering theorem, and transfinite induction will be used. The result presented below is a contribution to the general problem suggested above rather than to the theory of games. For the hypothesis of the theorem below precludes transitivity completely; that is, it precludes the existence of three elements a, b, c, such that a>b, b>c, and a>c. This restriction is too severe for the theory of games, just as is the assumption of transitivity. The problem remains open for weakly ordered systems which are not strictly acyclic but also do not satisfy the hypothesis of the theorem below.