Free objects and Gröbner-Shirshov bases in operated contexts
| Autor: | Zihao Qi, Guodong Zhou, Kai Wang, Yufei Qin |
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| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Polynomial Algebra and Number Theory Functor 13P10(Primary) 03C05 08B20 12H05 16S10 010102 general mathematics Mathematics - Rings and Algebras Basis (universal algebra) Type (model theory) 01 natural sciences Operator (computer programming) 0103 physical sciences Physics::Accelerator Physics Universal algebra 010307 mathematical physics Free object 0101 mathematics Algebraic number Mathematics |
| Zdroj: | Journal of Algebra. 584:89-124 |
| ISSN: | 0021-8693 |
| Popis: | 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 Gr\"obner-Shirshov basis of an algebra $A$ form a Gr\"obner-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. Comment: Slightly revised version of the published paper in Journal of Algebra |
| Databáze: | OpenAIRE |
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