Generalizations of the associative operad and convergent rewrite systems

Abstract

The associative operad is the quotient of the magmatic operad by the operad congruence identifying the two binary trees of degree $2$. We introduce here a generalization of the associative operad depending on a nonnegative integer $d$, called $d$-comb associative operad, as the quotient of the magmatic operad by the operad congruence identifying the left and the right comb binary trees of degree $d$. We study the case $d = 3$ and provide an orientation of its space of relations by using rewrite systems on trees and the Buchberger algorithm for operads to obtain a convergent rewrite system.