# Commutativity in double interchange semigroups

## TL;DR

This paper extends the understanding of commutativity in double interchange semigroups by analyzing relations with 10 arguments, using operad theory and geometric representations to establish new commutativity relations.

## Contribution

It introduces a framework for analyzing commutativity in DIS with 10 arguments using operads and geometric models, extending prior work.

## Key findings

- Established new commutativity relations for DIS with 10 arguments
- Developed algebraic and geometric representations of free DIS
- Proved theorems using diagrammatic reasoning

## Abstract

We extend the work of Kock (2007) and Bremner & Madariaga (2016) on commutativity in double interchange semigroups (DIS) to relations with 10 arguments. Our methods involve the free symmetric operad generated by two binary operations with no symmetry, its quotient by the two associative laws, its quotient by the interchange law, and its quotient by all three laws. We also consider the geometric realization of free double interchange magmas by rectangular partitions of the unit square $I^2$. We define morphisms between these operads which allow us to represent elements of free DIS both algebraically as tree monomials and geometrically as rectangular partitions. With these morphisms we reason diagrammatically about free DIS and prove our new commutativity relations.