The associative-commutative spectrum of a binary operation

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.

23 pages

References in corpus (5)

• On associative spectra of operations
• Associative spectra of binary operations
• Associative spectra of graph algebras I. Foundations, undirected graphs, antiassociative graphs
• The Nonassociativity of the Double Minus Operation
• Associative spectra of graph algebras II. Satisfaction of bracketing identities, spectrum dichotomy