Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Gröbner-Shirshov Bases Theory for Trialgebras

Version 1 : Received: 17 April 2021 / Approved: 19 April 2021 / Online: 19 April 2021 (15:19:28 CEST)

How to cite: Huang, J.; Chen, Y. Gröbner-Shirshov Bases Theory for Trialgebras. Preprints 2021, 2021040503 (doi: 10.20944/preprints202104.0503.v1). Huang, J.; Chen, Y. Gröbner-Shirshov Bases Theory for Trialgebras. Preprints 2021, 2021040503 (doi: 10.20944/preprints202104.0503.v1).

Abstract

We establish a Gröbner-Shirshov bases theory for trialgebras and show that every ideal of a free trialgebra has a unique reduced Gröbner-Shirshov basis. As applications, we give a method for the construction of normal forms of elements of an arbitrary trisemigroup, in particular, A.V. Zhuchok’s (2019) normal forms of the free commutative trisemigroups are rediscovered and some normal forms of the free abelian trisemigroups are first constructed. Moreover, the Gelfand-Kirillov dimension of finitely generated free commutative trialgebra and free abelian trialgebra are calculated respectively.

Subject Areas

Gröbner-Shirshov basis; normal form; Gelfand-Kirillov dimension; trialgebra; trisemigroup

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