Construction of free commutative integro-differential algebras by the method of Gr\"{o}bner-Shirshov bases
Construction of free commutative integro-differential algebras by the method of Gr\"{o}bner-Shirshov bases
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DOI：
10.1142/s0219498813501600
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发表时间：
2013-02
期刊：
影响因子：
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通讯作者：
Xing Gao;Li Guo;Shanghua Zheng
中科院分区：
文献类型：
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作者：
Xing Gao;Li Guo;Shanghua Zheng
本文利用Gr\“obner-Shirshov基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阶自由可换微分Rota-Baxter代数的合成钻石引理。我们还得到了这些代数上的一个弱单项序，从而得到了集合上自由交换积分微分代数的Gr\"{o}bner-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.
In this paper, we construct a canonical linear basis for free commutative integro-differential algebras by applying the method of Gr\"obner-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 Gr\"{o}bner-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.