左交换代数的“自由定理”

The “Freedom Theorem” for Left-Commutative Algebras

设X是一个有限集,LC(X)表示由X生成的自由左交换代数;f∈LC(X),Id(f)表示LC(X)的由f所生成的理想.对于任意的h,是否存在一个算法可以判断出h∈Id(f)或h■Id(f)?为了研究这个问题,文中应用Grbner-Shirshov基理论的思想方法在自由左交换代数的线性基底上定义了一个良序,证明了这个良序保持运算,重写了由一个多项式所生成的自由左交换代数的理想的元素的表达式,从而证明了一个定义关系的左交换代数具有可解的字问题并得到了左交换代数的"自由定理".

Let X be a finite set and LC( X) be the free left-commutative algebra generated by X. Let Id( f) be the ideal of LC( X) generated by f where f ∈LC( X). For each h,the problem is whether there is an algorithm to decide h ∈Id( f) or h■Id( f). This problem is studied by using the approach of Grbner-Shirshov bases theory. A well ordering on a linear basis of free left commutative algebra is defined. It is proved that the ordering is compatible with the product. The word problem for left-commutative algebras with a single defining relation is solved and the"freedom theorem"for left-commutative algebras is obtained by rewiting the elements of the ideal of free left-commutative algebra generated by one polynomial.