Permutation of elements in double semigroups

TL;DR

This paper investigates the permutation properties in double semigroups with two associative operations, revealing a degree 9 lowest level for commutativity and expressing these properties through graph cycles and computer algebra.

Contribution

It introduces a graph-based framework to analyze permutation properties in double semigroups and identifies the minimal degree where commutativity arises.

Findings

Commutativity in double semigroups occurs at degree 9.

A graph cycle approach models permutation properties.

Computer algebra confirms the minimal degree for commutativity.

Abstract

Double semigroups have two associative operations $\circ, \bullet$ related by the interchange relation: $( a \bullet b ) \circ ( c \bullet d ) \equiv ( a \circ c ) \bullet ( b \circ d )$. Kock \cite{Kock2007} (2007) discovered a commutativity property in degree 16 for double semigroups: associativity and the interchange relation combine to produce permutations of elements. We show that such properties can be expressed in terms of cycles in directed graphs with edges labelled by permutations. We use computer algebra to show that 9 is the lowest degree for which commutativity occurs, and we give self-contained proofs of the commutativity properties in degree 9.