Free objects and Gröbner-Shirshov bases in operated contexts
Free objects and Gröbner-Shirshov bases in operated contexts
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DOI：
10.1016/j.jalgebra.2021.04.042
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发表时间：
2021-03
影响因子：
0.9
通讯作者：
Zihao Qi;Yufei Qin;Kai Wang;Guodong Zhou
中科院分区：
文献类型：
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作者：
Zihao Qi;Yufei Qin;Kai Wang;Guodong Zhou
本文研究了带有算子的代数对象，如运算幺半群、运算代数等，并明确构造了这些运算上下文中的各种自由对象函子。对于算子满足一组Φ关系的运算代数（通常称为运算多项式恒等式（aka。OPIs）），Guo利用泛代数定义了自由对象，称之为自由Φ-代数。研究了代数上的自由Φ-代数。本文给出了一个温和的充分条件，使得Φ与代数A的Gröbner-Shirshov基构成代数A上自由Φ-代数的Gröbner-Shirshov基.提供了充分的例子，这一条件成立，如所有的Rota-Baxter型OPI，一类微分型OPI，平均OPI和雷诺OPI。
This paper investigates algebraic objects equipped with an operator, such as operated monoids, operated algebras etc. Various free object functors in these operated contexts are explicitly constructed. For operated algebras whose operator satisfies a set Φ of relations (usually called operated polynomial identities (aka. OPIs)), Guo defined free objects, called free Φ-algebras, via universal algebra. Free Φ-algebras over algebras are studied in details. A mild sufficient condition is found such that Φ together with a Gröbner-Shirshov basis of an algebraAform a Gröbner-Shirshov basis of the free Φ-algebra over algebraAin the sense of Guo et al.. Ample examples for which this condition holds are provided, such as all Rota-Baxter type OPIs, a class of differential type OPIs, averaging OPIs and Reynolds OPI.