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Séminaire Lotharingien de Combinatoire, 89B.10 (2023), 12 pp.

# Jia Huang and Erkko Lehtonen

# 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 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.

Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.

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