操作上下文中的自由对象和Gröbner-Shirshov基
Free objects and Gröbner-Shirshov bases in operated contexts
摘要：
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。在这些运算的上下文中，明确地构造了各种自由对象函子。对于其算子满足一组关系（通常称为操作多项式恒等式（aka））的$Φ$的操作代数。OPIs）），郭文贵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Guo等人意义下代数$A$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。
英文摘要：
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 algebra $A$ form a Gröbner-Shirshov basis of the free $Φ$-algebra over algebra $A$ in 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
