Gröbner bases and dimension formulas for ternary partially associative operads

Abstract

Dotsenko and Vallette discovered an extension to nonsymmetric operads of Buchberger's algorithm for Gröbner bases of polynomial ideals. In the free nonsymmetric operad with one ternary operation $(\ast \ast \ast )$, we compute a Gröbner 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 Gröbner bases to calculate the dimension formulas for these operads.

13 pages