The associative-commutative spectrum of a binary operation

• Published in: Discrete mathematics Vol. 346; no. 10; p. 113535
• Main Authors: Huang, Jia, Lehtonen, Erkko
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2023
• Subjects: Associative spectrum; Associative-commutative spectrum; Binary operation; Tree; Binary operation; Tree; Associative spectrum; Associative-commutative spectrum
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2023.113535

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 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 operad with (resp. without) permutations 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.