Gr\"obner bases and dimension formulas for ternary partially associative operads

TL;DR

This paper extends Buchberger's algorithm to compute Gr"obner bases for ternary partially associative operads, enabling the calculation of their dimension formulas and advancing algebraic operad theory.

Contribution

It introduces a method to compute Gr"obner bases for ternary operads with partial associativity, providing new tools for their algebraic analysis.

Findings

Computed Gr"obner basis for the ideal of partial associativity

Derived dimension formulas for ternary partially associative operads

Extended the operadic Buchberger's algorithm to new cases

Abstract

Dotsenko and Vallette discovered an extension to nonsymmetric operads of Buchberger's algorithm for Gr\"obner bases of polynomial ideals. In the free nonsymmetric operad with one ternary operation $({\ast}{\ast}{\ast})$, we compute a Gr\"obner basis for the ideal generated by partial associativity $((abc)de) + (a(bcd)e) + (ab(cde)$. In the category of $\mathbb{Z}$-graded vector spaces with Koszul signs, the (homological) degree of $({\ast}{\ast}{\ast})$ may be even or odd. We use the Gr\"obner bases to calculate the dimension formulas for these operads.