Abstract
We establish a method of Gröbner–Shirshov bases for trialgebras and show that there is a unique reduced Gröbner–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.
Highlights
The notion of a trialgebra was introduced by Loday [1] and investigated in many papers
For an arbitrary monomial-centers ordering on the linear basis, there is a unique reduced Gröbner–Shirshov basis for every ideal of free trialgebra
We show that an analogous result holds for trialgebras
Summary
The notion of a trialgebra (trioid known as trisemigroup) was introduced by Loday [1] and investigated in many papers (see, for example, [1,2,3,4,5]). If all operations of a trialgebra (trioid) coincide, we obtain an associative algebra (semigroup). A trialgebra has one more operation than dialgebra, so difficulty in the proof of some critical lemmas increases naturally. The paper is organized as follows: in Section 2, we first recall the linear basis constructed by Loday and Ronco [1] of the free trialgebra. We apply the method of Gröbner–Shirshov bases for certain trialgebras and trisemigroups to obtain normal forms and their Gelfand–Kirillov dimensions. For an arbitrary set X, the triwords over X are defined inductively as follows:. Of all normal triwords over X forms a linear basis of the free trialgebra generated by X. and define [ε]∅ = ε.
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