Generalizations of the associative operad and convergent rewrite systems

TL;DR

This paper introduces a family of generalized associative operads based on comb binary trees, analyzes the case for d=3, and develops a convergent rewrite system using operad Buchberger algorithms.

Contribution

It defines the d-comb associative operad, explores the case d=3, and constructs a convergent rewrite system for it using advanced operad techniques.

Findings

Defined the d-comb associative operad as a quotient of the magmatic operad.

Provided a convergent rewrite system for the case d=3.

Applied the Buchberger algorithm to operads to ensure convergence.

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.