Free objects and Grobner-Shirshov bases in operated contexts

[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 Phi of relations (usually called operated polynomial identities (aka. OPIs)), Guo defined free objects, called free Phi-algebras, via universal algebra. Free Phi-algebras over algebras are studied in details. A mild sufficient condition is found such that Phi together with a Grobner-Shirshov basis of an algebra A form a Grobner-Shirshov basis of the free Phi-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. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-10-15

[关键词] Differential type OPIs;Free objects;Grobner-Shirshov bases;Operated algebras;Free operated algebras over algebras;Operated polynomial identities;Rota-Baxter type OPIs;Universal algebra