Submitted:
17 April 2021
Posted:
19 April 2021
You are already at the latest version
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.
Keywords:
Gröbner-Shirshov basis
; normal form
; Gelfand-Kirillov dimension
; trialgebra
; trisemigroup
Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.
