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 f\in LC(X). For any h, the problem is whether there is an algorithm to decide h\in Id(f) or h\notin 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.