As a fundamental notion, the free differential algebra on a set is concretely constructed as the polynomial algebra on the differential variables. Such a construction is not known for the more general notion of the free differential algebra on an algebra, from the left adjoint functor of the forgetful functor from differential algebras to algebras, instead of sets. In this paper we show that a generator-relation presentation of a base algebra can be extended to the free differential algebra on this base algebra. More precisely, a Gröbner-Shirshov basis property of the base algebra can be extended to the free differential algebra on this base algebra, allowing a Poincaré-Birkhoff-Witt type basis for these more general free differential algebras. Examples are provided as illustrations.

作为基本概念，集合上的自由微分代数具体构造为微分变量的多项式代数。对于代数上的自由微分代数的更一般概念，从健忘算子的左伴随函子，从微分代数到代数，而不是集合，这种构造是未知的。在本文中，我们表明基本代数的生成器关系表示可以扩展到该基本代数上的自由微分代数。更准确地说，基本代数的Gröbner-Shirshov基属性可以扩展到该基本代数上的自由微分代数，从而为这些更通用的自由微分代数提供Poincaré-Birkhoff-Witt类型基础。提供示例作为说明。