左交换代数的“自由定理”

设X是一个有限集,LC(X)表示由$X$生成的自由左交换代数;$fin LC(X)$, 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$hin Id(f)$或$hnotin Id(f)$?为了研究这个问题, 文中应用Gr{o}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. 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.

The ``Freedom Theorem of Left-Commutative Algebras

Let X be a finite set and LC(X)$ be the free left-commutative algebra induced by X. Let Id(f)$ be the ideal of LC(X) induced by f where $fin LC(X)$. For any $h$, the problem is whether there is an algorithm to decide $hin Id(f)$ or $hnotin Id(f)$. This problem is studied by using the approach of Grobner-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 and that the element of the ideal of free left-commutative algebra induced by one polynomial is rewritten. The word problem for left-commutative algebras with a single defining relation is solved and the ``freedom theorem for left-commutative algebras is obtained.