A new classification of algebraic identities for linear operators on associative algebras

Bremner, Murray R.; Elgendy, Hader A.

doi:10.1016/j.jalgebra.2022.01.003

Cited by 2 publications

(11 citation statements)

References 15 publications

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“…In the present paper, we prove that all OPIs classified in [22] are Gröbner-Shirshov in the framework of [21], via the method of Gröbner-Shirshov bases. In other words, we show that OPIs classified in [22] are "good" OPIs searched in Rota's Classification Problem. Our study is a partial answer of Rota's Classification Problem.…”

Section: History In Solving Rota's Classification Problemmentioning

confidence: 64%

“…Recently, a new approach to study Rota's Classification Problem was brought forward by Bremner et al, based on the rank of matrices from OPIs [22]. They obtained six OPIs with degree 2 and multiplicity 1, and eighteen OPIs and two parametrized families with degree 2 and multiplicity 2.…”

Section: History In Solving Rota's Classification Problemmentioning

confidence: 99%

“…We then review the notation of Gröbner-Shirshov OPIs. Section 3 is devoted to prove that all OPIs in [22] are Gröbner-Shirshov. Three monomial orders ≤ dt , ≤ db and ≤ Dl are recalled.…”

Section: Outline Of the Papermentioning

confidence: 99%

“…Three monomial orders ≤ dt , ≤ db and ≤ Dl are recalled. Based on these three monomial orders, we first prove that all OPIs classified in [22] of degree 2 and multiplicity 1 are Gröbner-Shirshov (Theorem 1). Then, we show that all OPIs in [22] of degree 2 and multiplicity 2 are Gröbner-Shirshov (Theorem 2).…”

Section: Outline Of the Papermentioning

confidence: 99%

“…Based on these three monomial orders, we first prove that all OPIs classified in [22] of degree 2 and multiplicity 1 are Gröbner-Shirshov (Theorem 1). Then, we show that all OPIs in [22] of degree 2 and multiplicity 2 are Gröbner-Shirshov (Theorem 2). In Section 4, we conclude the main results obtained in this paper and propose some further problems to study.…”

Section: Outline Of the Papermentioning

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: History In Solving Rota's Classification Problemmentioning

confidence: 64%

Section: History In Solving Rota's Classification Problemmentioning

confidence: 99%

Section: Outline Of the Papermentioning

confidence: 99%

Section: Outline Of the Papermentioning

confidence: 99%

Section: Outline Of the Papermentioning

confidence: 99%

See 3 more Smart Citations