用Gröbner-Shirshov基方法构造自由交换积分微分代数
Construction of free commutative integro-differential algebras by the method of Gröbner-Shirshov bases
摘要：
本文利用Gröbner-Shirshov基的方法构造了自由交换积分-微分代数的一个标准线性基.建立了$n$阶自由交换微分Rota-Baxter代数的合成钻石引理.我们还得到了这些代数上的弱单项式序，使我们能够获得自由交换积分-微分代数在一个集合上的Gröbner-Shirshov基。最后，我们将泛函导子的概念推广到具有任意权和生成集的自由微分代数，并由此构造了自由交换积分-微分代数的典范线性基。
英文摘要：
In this paper, we construct a canonical linear basis for free commutative integro-differential algebras by applying the method of Gröbner-Shirshov bases. We establish the Composition-Diamond Lemma for free commutative differential Rota-Baxter algebras of order $n$. We also obtain a weakly monomial order on these algebras, allowing us to obtain Gröbner-Shirshov bases for free commutative integro-differential algebras on a set. We finally generalize the concept of functional derivations to free differential algebras with arbitrary weight and generating sets from which to construct a canonical linear basis for free commutative integro-differential algebras.
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