Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases

Guo, Li; Sit, William Y.; Gao, Xing; Zheng, Shanghua

doi:10.1145/2768577.2768644

Cited by 2 publications

(5 citation statements)

References 36 publications

(66 reference statements)

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“…Proof. The Rota-Baxter OPI, Nijenhuis OPI, average OPI and inverse average OPI are Rota-Baxter type OPIs, which are, respectively, Gröbner-Shirshov on Y with respect to the monomial order ≤ db [20]. Further, the monomial OPIs are respectively Gröbner-Shirshov on Y with respect to the monomial orders ≤ db , ≤ Dl and ≤ dt .…”

Section: Remarkmentioning

confidence: 99%

“…Therein theories of Gröbner-Shirshov bases and rewriting systems were applied successfully. Another important class of OPIs, namely Rota-Baxter type, was systematically studied in [20]. As to be expected from comparing integral calculus with differential calculus in analysis, the Rota-Baxter type OPIs are more challenging than the differential counterpart.…”

Section: History In Solving Rota's Classification Problemmentioning

confidence: 99%

“…As to be expected from comparing integral calculus with differential calculus in analysis, the Rota-Baxter type OPIs are more challenging than the differential counterpart. In [20], Rota-Baxter type OPIs were also characterized by Gröbner-Shirshov bases and rewriting systems. An outstanding achievement of applications of Gröbner-Shirshov bases and rewriting systems in the characterization of differential type OPIs and Rota-Baxter type OPIs sheds a light to apply them to study general OPIs, which was carried out in [21].…”

Section: History In Solving Rota's Classification Problemmentioning

confidence: 99%

“…Next, we recall two monomial orders ≤ db [20] and ≤ Dl [27], which will be used frequently in the remainder of the paper. Let (Y, ≤) be a well-ordered set.…”

Section: Opis Of Degree 2 and Multiplicitymentioning

confidence: 99%

“…Here, u * i ≤ db v * i and u i ≤ db v i are compared by the induction hypothesis. Then, ≤ db is a monomial order on S(Y) [20].…”

Section: Opis Of Degree 2 and Multiplicitymentioning

confidence: 99%

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New Operated Polynomial Identities and Gröbner-Shirshov Bases

2022

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Twenty years ago, Rota posed the problem of finding all possible algebraic identities that can be satisfied by a linear operator on an algebra, named Rota’s Classification Problem later. Rota’s Classification Problem has proceeded two steps to understand it and has been studied actively recently. In particular, the method of Gröbner-Shirshov bases has been used successfully in the study of Rota’s Classification Problem. Quite recently, a new approach introduced to Rota’s Classification Problem and classified some (new) operated polynomial identities. In this paper, we prove that all operated polynomial identities classified via this new approach are Gröbner-Shirshov. This gives a partial answer of Rota’s Classification Problem.

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Section: Remarkmentioning

confidence: 99%

Section: History In Solving Rota's Classification Problemmentioning

confidence: 99%

Section: History In Solving Rota's Classification Problemmentioning

confidence: 99%

“…Next, we recall two monomial orders ≤ db [20] and ≤ Dl [27], which will be used frequently in the remainder of the paper. Let (Y, ≤) be a well-ordered set.…”

Section: Opis Of Degree 2 and Multiplicitymentioning

confidence: 99%

“…Here, u * i ≤ db v * i and u i ≤ db v i are compared by the induction hypothesis. Then, ≤ db is a monomial order on S(Y) [20].…”

Section: Opis Of Degree 2 and Multiplicitymentioning

confidence: 99%

See 3 more Smart Citations