格序群的一个特征性质

A characteristic Property of Lattice-ordered group

本文讨论了格序群的一个特征性质,回答了G.Birkhoff一篇论文(Birkhoff.G, Lattice-orderred gronps, Ann, of Math. Vol.43, NO.2(1942)298—331)第331页中的下述问题: 问题15,求对于定理9内条件(C)的一个更直接的代替:在运算上的一个简单条件或简单条件的集,必须和充分的使运算(a-b)~*+b是结合的。 本文给出了上述问题的解答,即定理中的条件(Ⅳ)。 注意:在BirkhoffG.一文的第302页有下述定理。 定理9 一个l—群G,可被定义为一个群,具有一个一元运算,集在所有内自同构之下不变,且适合: (a)0~*=0 (b)c=c~*-(-c)~* (c)运算a∨b≡(a-b)~*=b是结合的。

This paper discusses a characteristic property of lattice-ordered group, answering the problem 15of a paper (Birkhoff.G, Latlive-ordered groups. Ann. of math 1942, 43, (2) 298-331) .In page 331 of this paper have the following Problem.Problem 15. Find a more direct snbstitute for condition (c) in theorem 9: i. e., a Simpcl condition or set of Simple conditions on the operation a a necessary and snfficient to make the operation (a-b) * + b associative.This paper give a Solution of this Problem, namely, the condition (Ⅳ) in the theorem.Remark In page 302 of Birkhoff's paper have the following theorem.Tbeorem 9.An 1-group may be defined as a group with a unary operation a a*, which is invariant under all inner automorphisms and satisfies:(Ⅰ) 0*=0(Ⅱ) c=c*- (-c) *,(Ⅲ) the binary operation (a-b) *+b is associative.