The associative-commutative spectrum of a binary operation
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866916498751094784 |
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| author | Huang, Jia Lehtonen, Erkko |
| author_facet | Huang, Jia Lehtonen, Erkko |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_11826 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The associative-commutative spectrum of a binary operation Huang, Jia Lehtonen, Erkko Combinatorics Rings and Algebras 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. |
| title | The associative-commutative spectrum of a binary operation |
| topic | Combinatorics Rings and Algebras |
| url | https://arxiv.org/abs/2202.11826 |