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arXiv · 2202.11826

The associative-commutative spectrum of a binary operation

Jia Huang·Erkko Lehtonen
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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.

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Jia Huang, Erkko Lehtonen. 2022-02-23. The associative-commutative spectrum of a binary operation. https://doi.org/10.1016/j.disc.2023.113535

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