The associative-commutative spectrum of a binary operation

TL;DR

This paper introduces the associative-commutative spectrum as a quantitative measure of how much a binary operation deviates from being both associative and commutative, extending previous measures of nonassociativity.

Contribution

It defines and studies the associative-commutative spectrum, providing general results, explicit calculations for specific operations, and proposing future research directions.

Findings

Defined the associative-commutative spectrum as a cardinality of a symmetric operad.

Calculated the spectrum for certain binary operations.

Connected the spectrum to the structure of operads derived from groupoids.

Abstract

We initiate the study of a quantitative measure for the failure of a binary operation to be commutative and associative. We call this measure the associative-commutative spectrum as it extends the so-called associative spectrum (also known as the subassociativity type), which measures the nonassociativity of a binary operation. In fact, the associative-commutative spectrum (resp. associative spectrum) is the cardinality of the symmetric (resp. nonsymmetric) operad obtained naturally from a groupoid (a set with a binary operation). In this paper we provide some general results on the associative-commutative spectrum, precisely determine this measure for certain binary operations, and propose some problems for future study.