Gröbner–Shirshov Bases Theory for Trialgebras

We establish a method of Grobner–Shirshov bases for trialgebras and show that there is a unique reduced Grobner–Shirshov basis for every ideal of a free trialgebra. 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.